The company's decision to engage with Mr Musk, taken earlier on Sunday, does not mean that it will accept his $54.20 a share bid, the sources said. It signalled, however, that Twitter was exploring whether a sale to Mr Musk was possible on attractive terms.
Mr Musk, chief executive of electric car maker Tesla, has been meeting with Twitter shareholders in the past few days seeking support for his bid. He has said Twitter needs to be taken private to grow and become a genuine platform for free speech.
Many Twitter shareholders reached out to the company after Mr Musk outlined a detailed financing plan for his bid on Thursday and urged it not to let the opportunity for a deal slip away, Reuters reported earlier on Sunday.
Mr Musk's insistence that his bid for Twitter is his "best and final" offer has emerged as a hurdle in the deal negotiations, the sources said. Nevertheless, Twitter's board has decided to engage with Mr Musk to gather more information on his ability to complete the deal, and potentially get better terms, the sources added.
Twitter has not yet decided if it will explore a sale to put pressure on Mr Musk to raise his bid, according to the sources.
Twitter wants to know more about any active investigations by regulators into Mr Musk, including by the US Securities and Exchange Commission (SEC), that would present a risk to the deal being completed, one of the sources said.
Securities lawyers say that Mr Musk, who settled charges that he misled investors by suggesting four years ago he had secured funding to take Tesla private, may have breached SEC disclosure rules as he amassed a stake in Twitter earlier this year.
Twitter is also looking into whether regulators in any of the major markets it operates would object to Mr Musk owning the company, the source added. Were Twitter to establish that a sale to Mr Musk would be risky, it could ask for a sizeable break-up fee, according to the sources.
The social media company adopted a poison pill after Mr Musk made his offer to prevent him from raising his more than 9 per cent stake in the company above 15 per cent without negotiating a deal with its board. In response, Mr Musk has threatened to launch a tender offer that he could use to register Twitter shareholder support for his bid.
A concern that Twitter's board weighed was that unless it sought to negotiate a deal with Mr Musk, many shareholders could back him in a tender offer, the sources said.
While the poison pill would prevent Twitter shareholders from tendering their shares, the company is worried that its negotiating hand would weaken considerably if it was shown to be going against the will of many of its investors, the sources added.
Representatives for Twitter and Mr Musk did not immediately respond to requests for comment.
The Wall Street Journal reported earlier on Sunday that Mr Musk and Twitter would meet to discuss the acquisition offer.
The price expectations among Twitter shareholders for the deal diverge largely based on their investment strategy, the sources said.
Active long-term shareholders, who together with index funds hold the biggest chunk of Twitter shares, have higher price expectations, some in the $60-per-share range, the sources said. They are also more inclined to give Parag Agrawal, who became Twitter's chief executive in November, more time to boost the value of the company's stock, the sources added.
"I don't believe that the proposed offer by Elon Musk comes close to the intrinsic value of Twitter given its growth prospects," Saudi Arabia's Prince Alwaleed bin Talal, a Twitter shareholder, tweeted on April 14.
Short term-minded investors such as hedge funds want Twitter to accept Mr Musk's offer or ask for only a small increase, the sources said. Some of these are fretting that a recent plunge in the value of technology stocks amid concerns over inflation and an economic slowdown makes it unlikely Twitter will be able to deliver more value for itself anytime soon, the sources added.
"I would say, take the $54.20 a share and be done with it," said Sahm Adrangi, portfolio manager at Kerrisdale Capital Management, a hedge fund that owns 1.13 million shares in Twitter, or 0.15 per cent of the company, and has been an investor since early 2020.
One silver lining for Twitter's board is that Mr Musk's offer did not appear to convert much of his army of 83 million Twitter followers into new shareholders in the San Francisco-based company who could back his bid, the sources said.
Twitter's retail investor base has increased from about 20 per cent before Mr Musk unveiled his stake on April 4 to some 22 per cent, according to the sources.