Mr. Richardson said on CNN on Wednesday that he believed Mr. Biden was moved by the humanitarian need to secure Mr. Reed’s release. Family members said he was suffering from active tuberculosis while in detention in Russia.

But Mr. Biden’s decision to approve the prisoner swap also came after weeks of intense public pressure from Mr. Reed’s parents, who initially criticized the president for refusing to meet with them about their son’s case.

Mr. Biden called them on the phone in early March after the couple stood on the side of the road in Texas and waved as the presidential motorcade drove by on the way to an event about veterans. Three weeks later, in late March, Mr. Biden invited the couple into the White House for a brief conversation after they protested outside the building.

That public pressure gained momentum as the war in Ukraine intensified, raising concerns that the deteriorating relationship between the United States and Russia would make a breakthrough less likely. In fact, American officials and Mr. Richardson said the fighting in Ukraine did not appear to hamper Russia’s willingness to discuss the swap.

Mr. Reed was initially detained by Russian authorities in August 2019. His family had increasingly expressed concern about his health while he was in detention, and footage aired on Wednesday by Russian state-run television showed what appeared to be a visibly gaunt Mr. Reed being escorted to a Russian plane at Moscow’s Vnukovo airport. Mr. Reed’s father, Joey Reed, told CNN that his son had been flown to Turkey, where the exchange took place.

A native of Texas, Mr. Reed traveled to Russia in May 2019 to visit his Russian girlfriend, whom he had met on a dating website, and to take language lessons. One week before his planned return to the United States, he went to a party at a park outside Moscow, where he drank extensive amounts of vodka.

Mr. Reed then got agitated, prompting his girlfriend and friends to call the police. Officers who arrived at the scene decided to take him to a police station, where he was interviewed by Russian security agents. Shortly after, he was accused of assaulting and endangering the lives of the two police officers who had driven him there.