He added that being public "puts enormous pressure" on Tesla, forcing it to make decisions "not necessarily right for the long-term".
Musk's tweets helped push Tesla's stock price more than 13 percent above the previous day's close. He also said he had secured funding for the proposal, without providing details.
Short-sellers make a profit by borrowing overpriced shares, selling them and then buying them back at what they hope will be a lower price.
Isaacs claims that Tesla's volatility on Wall Street after the Twitter storm has cost shareholders shorting the stock hundreds of millions of dollars, and caused all Tesla securities purchasers to pay inflated prices over the three-day period leading up to Friday 10 August. They are canvassing investors including large asset managers, the people said.
In one of the lawsuits, the plaintiff Kalman Isaacs said Musk's tweets were false and misleading, and together with Tesla's failure to correct them amounted to a "nuclear attack" created to "completely decimate" short-sellers.
Tesla's share price edged higher as markets closed late last night, increasing 0.86 per cent to $355.49.
Ratings agency Moody's also said on Thursday that Tesla's consideration of going private based on Musk's letter to shareholders published after his tweets on Tuesday was negative for the company's credit outlook.
Tesla's board has allegedly not yet received comprehensive information regarding the finances required to take the company private, Reuters reports, citing sources familiar with the matter. In a barrage of new tweets, he joked about "short shorts" being included in "Tesla merch", suggested several potential names for the "shorts", and asked his followers which "short shorts" are their favorite.
The US Securities and Exchange Commission is understood to have already opened an inquiry into Mr Musk's tweets.