The case alleges he violated California corporate laws in a number of ways.
It accuses the Tesla boss of “wrongful conduct” as his “false statements and market manipulation have created ‘chaos’ at Twitter’s headquarters in San Francisco”.
Twitter shares are around 27% lower than Mr Musk’s $54.20 offer price.
The proposed class action lawsuit was filed this week at the US District Court for the Northern District of California by investor William Heresniak, who said he was acting “on behalf of himself and all others similarly situated”.
A class-action is a lawsuit that has been filed or is defended by an individual acting on behalf of a group of people.
The lawsuit claimed Mr Musk benefitted financially by delaying the disclosure of his significant stake in Twitter, and his plan to become a board member of the company.
It also claimed that several tweets posted by Mr Musk, who is a regular Twitter user with more than 95m followers, were “misleading”.
It included a post in which Mr Musk saidhis takeover bid for the social media firm was on holdbecause of his doubts over the number of fake accounts on the platform.
The tweet on 13 May “constituted an effort to manipulate the market for Twitter shares as he knew about the fake accounts,” the lawsuit said.
View original tweet on Twitter
It also said Mr Musk “doubled down” on his allegations four days later, by stating on Twitter that the deal “cannot go forward”.
On Friday, Frank Bottini, who is one of the lawyers representing the Twitter investors, told the BBC that the lawsuit was filed as Mr Musk “continues to disparage the company he wants to buy for $44bn in an effort to renegotiate the purchase price”.
“The complaint we filed in San Francisco seeks to hold Musk liable for his unlawful conduct,” Mr Bottini said.
Mr Musk’s lawyers and Tesla did not immediately respond to a BBC request for comment on Friday.
Twitter declined to comment when contacted by the BBC.
Analysts have speculated that Mr Musk may be looking for ways to lower his takeover offer or walk away from the deal.
He has tweeted several times that he was concerned about the number of fake accounts, or bots, on Twitter.
A bot is a software programme that sends out automated posts and is often associated with misinformation on social media platforms. Mr Musk has also hinted that he may seek to pay less for Twitter than the $44bn agreed with the company’s board in March.
Speaking at a technology conference earlier this month, he said striking a deal at a lower price was “not out of the question”.
Earlier this month, a Florida pension fund also challenged Mr Musk’s move to buy Twitter as it claimed a deal could not be struck in months as planned.
The Orlando Police Pension Fund said Mr Musk was an “interested shareholder” in Twitter, as he had made agreements with major shareholders, including its co-founder Jack Dorsey, before he offered to buy the business. Citing the law in the US state of Delaware, where Twitter is incorporated, it said the deal should not be allowed to close before 2025.
The fund did not disclose its stake in the social media platform, although it said it has been “a beneficial owner of Twitter common stock” at “all relevant times”.
Source: BBC