WASHINGTON — The Biden administration is vigorously debating how much the United States can or should assist an investigation into Russian atrocities in Ukraine by the International Criminal Court in The Hague, according to officials familiar with internal deliberations.

The Biden team strongly wants to see President Vladimir V. Putin of Russia and others in his military chain of command held to account. And many are said to consider the court — which was created by a global treaty two decades ago as a venue for prosecuting war crimes, crimes against humanity and genocide — the body most capable of achieving that.

But laws from 1999 and 2002, enacted by lawmakers wary that the court might investigate Americans, limit the government’s ability to provide support. And the United States has long objected to any exercise of jurisdiction by the court over citizens of countries that are not part of the treaty that created it — like the United States, but also Russia.

The internal debate, described by senior administration officials and others familiar with the matter on the condition of anonymity, has been partly shaped by a previously undisclosed 2010 memo by the Justice Department’s Office of Legal Counsel. Obtained by The New York Times, the memo interprets the scope and limits of permissible cooperation with the court.

The discussions have also been marked by Pentagon opposition to softening the U.S. stance, even as congressional Republicans, long skeptics of the court, have signaled openness to finding a way to help it bring Russian officials to justice.

For now, officials said, the primary focus has been on compiling evidence of apparent war crimes that are still unfolding — both the details of particular killings and intelligence that President Biden’s national security adviser, Jake Sullivan, asserted on Sunday indicates a high-level plan to brutalize the civilian population into terrorized subjugation.

“This was something that was planned,” he said on ABC’s “This Week,” adding, “Make no mistake, the larger issue of broad-scale war crimes and atrocities in Ukraine lies at the feet of the Kremlin and lies at the feet of the Russian president.”

But the unresolved deliberations about where to channel such intelligence explain why administration officials have been hazy about where efforts to prosecute Russian war crimes should center — even as evidence of large-scale atrocities has mounted, prompting Mr. Biden to label Mr. Putin a “war criminal” and to call for a “war crimes trial.”

Mr. Sullivan, was vague, for example, at a news briefing last week when a reporter asked whether the president envisioned such a prosecution playing out at the International Criminal Court or some other tribunal.

“We have to consult with our allies and partners on what makes most sense as a mechanism moving forward,” he said. “Obviously, the I.C.C. is one venue where war crimes have been tried in the past, but there have been other examples in other conflicts of other mechanisms being set up.”

But setting up other venues would raise its own obstacles. Among them, while the United Nations Security Council in the past helped establish special international courts to handle conflicts in places like Rwanda and the former Yugoslavia, Russia can veto any Council resolution seeking to establish a tribunal for Ukraine.

There are reasons to doubt that Mr. Putin and other senior Kremlin officials responsible for the war may ever stand trial, so long as they remain in power and ensconced in Russia. Still, war-crimes indictments, human rights specialists say, serve a “naming and shaming” function even without trials — and can inhibit defendants’ ability to travel abroad.

Another possibility is a nation’s court with jurisdiction over war crimes on Ukrainian soil. Germany, for example, has war-crimes and crimes-against-humanity laws that cover the world. Prosecutors there said in March that they had started gathering evidence of deliberate attacks on civilians and civilian infrastructure, and two former ministers filed a complaint there last week asking prosecutors to charge Russian officials.

Ukraine’s own prosecutor general has asked for international help in gathering evidence. Attorney General Merrick B. Garland said in recent days that administration officials were working on a multinational effort to shore up Ukraine’s efforts, while also holding discussions with European counterparts.

Still, with Ukraine under continuing assault, the capacity of its justice system may be limited. The International Criminal Court, by contrast, is already set up — and it specializes in conducting this very kind of investigation and prosecution.

Against that backdrop, the State Department has said that the United States “welcomed the fact” that the court has opened an investigation into the war in Ukraine, and Biden administration officials are weighing what the United States can do to help it.

One set of issues is primarily legal. A group of top national security lawyers across the administration has been wrestling with how to navigate the limits imposed by a pair of laws that Congress enacted a generation ago. Those laws curtail the aid the American government may provide to the court, but are ambiguous in places.

The deliberations have centered on a 26-page opinion by the Office of Legal Counsel that interpreted those laws for the executive branch.

The memo looked at the kinds of assistance that the United States had offered to the tribunals for war crimes and genocide in the former Yugoslavia and Rwanda, which proved crucial to making them work — like sending dozens of experienced Justice Department prosecutors and contributing more than $500 million to cover operational costs.

But a 1999 appropriations law bars the government from spending funds to support the International Criminal Court. The memo concludes that Congress banned both donating money to the court directly and donating material items, like supplying a computer system or building a courthouse — and that the law permits no exceptions.

The memo also analyzes a 2002 law, the American Servicemembers Protection Act. It bars giving the court other kinds of support — like sharing intelligence, training its staff or lending it personnel. The memo concludes that the United States cannot offer general institutional support, but can provide such help for “particular cases.”

Unlike the funding ban, the 2002 law permits “rendering assistance to international efforts to bring to justice” a list of offenders from that era, like Saddam Hussein and Osama bin Laden, along with any other foreign citizens who are accused of genocide, war crimes or crimes against humanity.

Even as administration lawyers struggle with how much wiggle room the government has as it tries to hold Russia accountable, there are signs of bipartisan interest in Congress in potentially rescinding or modifying those laws so the United States can more broadly help the court.

Last month, the Senate unanimously passed a resolution by Senator Lindsey Graham, Republican of South Carolina, backing any investigation into war crimes committed by Russian forces and proxies. It praised the International Criminal Court and encouraged “member states to petition the I.C.C.” to investigate and prosecute Russian atrocities — as at least 41 nations have done.

Mr. Graham has since been working with Senator Richard J. Durbin of Illinois, the No. 2 Democrat in the Senate, on what they hope will be fast-track, bipartisan legislation responding to outrage over events in Ukraine.

Their initial focus, Mr. Graham said in an interview, has been on developing legislative language to expand the domestic War Crimes Act so that American courts would have jurisdiction over such crimes by noncitizens abroad.

But Mr. Graham said that the International Criminal Court appeared “to be the only venue that works” for investigating Mr. Putin. He said Congress would also “look at the laws on the books and see if they need to be changed to make sure these investigations can be supported, either financially or to provide any intelligence or manpower.”

A related issue under discussion among administration officials is whether the United States should soften its longstanding objection to the court exercising jurisdiction over citizens from a country that is not a party to its treaty, according to officials.

On the table is whether those decisions should instead depend on whether a particular country has a functioning justice system that can handle allegations of war crimes. The rationale is that it would be legitimate for the court to investigate Russian war crimes because Mr. Putin and his commanders appear to be committing them with domestic impunity.

Pentagon officials, however, are said to be balking. They contended that moving to a case-by-case approach would be shortsighted because it would make it harder for the United States to argue against court investigations into potential war crimes by American forces, officials said.

Some opponents of changing the American position are also said to have pointed to Israel — an ally that is also not a party to the treaty. The United States has objected to an investigation by the court of potential war crimes by Israeli forces.

But calls for shifting the U.S. position are growing louder.

Mr. Graham asserted that the court was set up to deal with situations only where the rule of law has collapsed — unlike the court systems in the United States or Israel, he said. By contrast, he argued, “there is no rule of law in Russia any more than there would be in certain parts of Africa” where the court has prosecuted warlords for atrocities.

And in a Washington Post opinion column last week, John Bellinger, a national security lawyer in the George W. Bush administration, and Christopher J. Dodd, a former Democratic senator who was responsible for adding the exception to the 2002 law, argued that “U.S. support for an I.C.C. investigation of Russian war crimes would not constitute a double standard or be inconsistent with U.S. objections to the court’s claimed jurisdiction over U.S. personnel.”

While most of the world’s democracies joined the court a generation ago — including close U.S. allies like Britain — many American leaders were wary, fearing that it could be used or misused someday to prosecute American forces.

In 2000, President Bill Clinton signed the 1998 treaty creating the court, known as the Rome Statute, but he also called it flawed and never submitted it to the Senate for ratification. Two years later, Mr. Bush essentially withdrew that signature.

Still, by 2008, Mr. Bellinger — then the top State Department lawyer — declared that the United States accepted the “reality” of the court, acknowledging that it “enjoys a large body of international support.” The Obama administration bolstered its efforts to prosecute warlords in Africa, offering rewards for the capture of fugitives indicted by the court.

Relations plunged during the Trump administration, when a top prosecutor for the court tried to investigate the torture of terrorism detainees during the Bush administration. The government imposed sanctions on court personnel, and Secretary of State Mike Pompeo denounced it as corrupt.

But in 2021, the Biden administration revoked those sanctions and a new top prosecutor dropped the investigation.

In light of that history, the unanimous vote for the Senate resolution supporting the International Criminal Court represented a striking change. Mr. Graham attributed that shift to the “war crimes spree” by Mr. Putin.

“I would say this is one of Putin’s bigger accomplishments,” Mr. Graham said. “I didn’t think it was possible but he did it — and that’s for him to rehabilitate the I.C.C. in the eyes of the Republican Party and the American people.”