The founder of the electric carmaker also indicated that he would need to raise far less than the $70bn it has been estimated he would need.
In a blog post, Mr Musk said that he only wanted to buy out shareholders who no longer wanted to own Tesla shares.
The announcement comes just days after investors filed a lawsuit claiming that he misled the market.
Mr Musk announced on Twitter on 7 August that he was considering taking Tesla private, adding "funding secured".
Since then he has been facing questions about where he would obtain the funding for his proposed $420 a share offer.
Shares in Tesla rose slightly to $357 in morning trading in New York.
They have fallen back from the highs near $380 they touched after his initial tweet. The investors suing Tesla are short-sellers, who bet on share prices falling and claim to have lost millions as a result of Mr Musk's comments.
On Monday, the Tesla founder said he had met with the Saudi sovereign wealth fund after it bought a 5% stake in the company.
He left that meeting on 31 July "with no question that a deal ... could be closed - it was just matter of getting the process moving".
The fund was "eager to proceed" with financing "a going private transaction" after that meeting, Mr Musk said.
The Saudi fund had approached him "multiple times" since the start of last year about taking the company private, Mr Musk said.
"They first met with me at the beginning of 2017 to express this interest because of the important need to diversify away from oil," he said.
Mr Musk, who owns almost 20% of Tesla, had sent the tweet so he could "reach out" to the company's largest investors to see they were interested in backing a deal.
Since then, he said he had kept in touch with the Saudi fund and that its managing director had "expressed support" for proceeding with the deal, subject to legal checks.
Mr Musk said he would not need $70bn to buy out all shareholders as he only intended to offer $420 a share to those who did not want to remain investors if the company did go private.
"My best estimate right now is that approximately two thirds of shares owned by all current investors would roll over into a private Tesla," he said.
The plan means Mr Musk would not need to borrow huge sums to pull off the deal.
He has said that de-listing the company from the stock market would avoid pressure to meet quarterly financial targets and end "negative propaganda" from short-sellers.
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