Twitter may be moving closer to a deal with Elon Musk.

The board of the social media service met on Sunday morning to discuss Mr. Musk’s unsolicited $46.5 billion bid to buy the company, after he began lining up financing for his offer last week, two people with knowledge of the situation said. The financing was a turning point for how Twitter’s board viewed Mr. Musk’s bid of $54.20 a share, enabling the company’s 11 board members to seriously consider his offer, the people said.

Twitter’s board planned to meet with Mr. Musk’s side later on Sunday to discuss other contours around a potential deal, said the people, who spoke on the condition of anonymity because they were not authorized to discuss confidential information. Those details include a timeline to close any potential deal and any fees that would be paid if an agreement was signed and then fell apart.

Any deal remains far from certain, but the willingness of Twitter’s board to engage with Mr. Musk, the world’s richest man, represents a step forward. Mr. Musk, who has more than 83 million followers on Twitter and began amassing shares in the company earlier this year, declared his intent to buy the company on April 14 and take it private. But his proposal was quickly dismissed by Wall Street because it was unclear if he could come up with the money to do the deal. Twitter also adopted a “poison pill,” a defensive maneuver that would prevent Mr. Musk from accumulating more of the company’s stock.

Mr. Musk updated his proposal last week, putting pressure on Twitter to more seriously consider his bid. In a securities filing that was made public on Thursday, Mr. Musk detailed how he had put together financing from the investment bank Morgan Stanley and a group of other lenders, which were offering $13 billion in debt financing, plus another $12.5 billion in loans against his stock in Tesla, the electric carmaker that he runs. He was expected to add about $21 billion in equity financing.

Now what had appeared to be a highly improbable deal may be more probable. The situation involving Twitter and Mr. Musk remains fluid and fast-moving, the people with knowledge of the situation said.

A Twitter spokesman declined to comment. In previous public statements, the company had said its board was “continuing to conduct a careful, comprehensive and deliberate review to determine the course of action in the best interest of the company and all Twitter stockholders.”

Mr. Musk did not respond to a request for comment. The Wall Street Journal earlier reported Twitter’s increased receptivity to Mr. Musk’s bid.

Wall Street was likely to view the openness of Twitter’s board to Mr. Musk’s bid as “the beginning of the end for Twitter as a public company with Musk likely now on a path to acquire the company unless a second bidder comes into the mix,” Dan Ives, an analyst at Wedbush Securities, wrote in a note on Sunday.

Mr. Musk’s offer for Twitter is a 54 percent premium over the share price the day before he began investing in the company in late January. But Twitter’s shares traded higher than Mr. Musk’s bid for much of last year.

Several analysts have said they expected Twitter’s board to only accept a bid that valued it at a minimum of $60 a share. Twitter’s stock rose above $70 a share last year when the company announced goals to double its revenue, but has since fallen to around $48 as investors have questioned its ability to meet those targets.

Mr. Musk, 50, has made clear that he sees many deficiencies in Twitter as a social media service. He has said that he wants to “transform” the company as a “platform for free speech around the globe” and that it requires vast improvements in its product and policies.

Mr. Musk has tried to negotiate with Twitter using the service itself, threatening in several tweets that he might take his bid directly to the company’s shareholders in what is called a “tender offer.” A tender offer is a hostile maneuver in which an outside party circumvents a company’s board by asking shareholders to sell their shares directly to them.

He has also acted erratically on the platform, raising concerns over how he might manage the service should he be in charge of it. On Saturday, Mr. Musk took aim at the billionaire Bill Gates, saying that Mr. Gates had taken a “short” position on the stock of Tesla, which meant that Mr. Gates was betting the carmaker’s shares would fall. On Sunday, Mr. Musk tweeted that he was “moving on” from making fun of Mr. Gates.

Even so, Mr. Musk maintains amicable ties with some high-ranking members of Twitter. Over the weekend, Mr. Musk traded friendly tweets with Jack Dorsey, the company’s co-founder and a board member. Mr. Dorsey stepped down as Twitter’s chief executive in November and soon will be leaving its board.

Both men share similar views on cryptocurrencies and on promoting more free speech online. When Mr. Musk briefly flirted with joining Twitter’s board this month, Mr. Dorsey tweeted, “I’m really happy Elon is joining the Twitter board! He cares deeply about our world and Twitter’s role in it.”

On Friday, Block, a financial services company led by Mr. Dorsey, disclosed that he had changed his title at the firm from chief executive to “Block Head.” That shift appeared to resonate with Mr. Musk.

“Your new title at Block is fire,” Mr. Musk tweeted at Mr. Dorsey on Saturday, using two flame emojis to signify his approval. Last year, Mr. Musk had changed his title at Tesla from chief executive to technoking.