In a surprise announcement, Musk said he would pay $54.20 a share for Twitter, valuing it at about $41b.
It recently emerged that Musk was Twitter's biggest shareholder after he built up a large stake in the firm.
He said that if his offer was not accepted: "I would need to reconsider my position as a shareholder".
A filing with the US financial regulator appeared to show text and/or voice messages from Musk to Twitter's board, showing that he had raised the idea at the weekend that the business should go private.
Musk had been invited to join the board, but Twitter announced on Sunday he had decided against it.
In the messages, Musk said he was not "playing the back-and-forth game" and insisted its shareholders would "love the high price" he was offering for the firm.
In the filing with the US financial regulator Musk said: "I invested in Twitter as 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.
"However, since making my investment I now realize the company will neither thrive nor serve this societal imperative in its current form. Twitter needs to be transformed as a private company."
He added: "Twitter has extraordinary potential. I will unlock it."
Twitter had offered Musk a seat on its board after it emerged that he held a 9.2 percent stake in the company.
The role was set to become effective last weekend but then Musk decided against taking it.
On Monday, Twitter's chief executive Parag Agrawal had tweeted that he thought the decision by Musk not take a seat on the board was "for the best".
He added: "We have and will always value input from our shareholders whether they are on our board or not. Elon is our biggest shareholder and we will remain open to his input."
Though he also warned staff: "There will be distractions ahead."
After that announcement, Musk posted a cryptic tweet of smiling emoji with its hand over its mouth.