Should Twitter put such a measure in place, Mr. Musk could take his bid directly to shareholders with a so-called tender offer. If they like Mr. Musk’s offer, they could sell their stock to the billionaire, allowing him to gain control of the company.

The board is also considering soliciting offers from other companies that may want to acquire Twitter, one of the people familiar with its plans said, another common move among takeover targets. It was not clear if Twitter had approached anyone yet.

A deal could require Mr. Musk to obtain $15 billion to $20 billion in debt financing, said Dan Ives, an analyst at Wedbush Securities. Though Mr. Musk is the wealthiest man in the world, much of his worth is tied up in Tesla shares.

Twitter could question Mr. Musk’s ability to finance the deal or claim it undervalues the company. The offer is a 54 percent premium over the share price the day before he began investing in the company in late January. But shares of Twitter traded higher than Mr. Musk’s bid for much of last year.

When Mr. Musk announced his stake in Twitter, several board members welcomed him, including Twitter’s co-founder Jack Dorsey. Twitter offered Mr. Musk a seat in its boardroom, a move that executives hoped would prevent him from publicly criticizing the company. But the welcome quickly soured. Mr. Musk turned down the board seat.

By Wednesday, Mr. Musk was ready to disclose his next move. In a message to Bret Taylor, the chairman of Twitter’s board, Mr. Musk said: “I believe in its potential to be the platform for free speech around the globe, and I believe free speech is a societal imperative for a functioning democracy,” according to a filing with the Securities and Exchange Commission that was later made public. “Twitter needs to be transformed as a private company.”

Mr. Taylor, a co-chief executive of Salesforce, has served on Twitter’s board since 2016, when Salesforce considered buying Twitter. Mr. Taylor joined Salesforce a year later, after it acquired his own company, Quip.