Media Matters said earlier this week that corporate advertisements were being placed alongside anti-Semitic content, including praise for both Adolf Hitler and the Nazis.
Apple, Disney and IBM are among the firms to have suspended all advertising with the social media site.
Lionsgate Entertainment, the film studio behind such movies as La La Land, has also pulled advertising, Bloomberg reported.
“The split second court opens on Monday, X Corp will be filing a thermonuclear lawsuit against Media Matters and ALL those who colluded in this fraudulent attack on our company,” Musk said in a post on X, formerly known as Twitter.
“This week Media Matters for America posted a story that completely misrepresented the real experience on X, in another attempt to undermine freedom of speech and mislead advertisers,” a statement posted by Musk said.
“Above everything, including profit, X works to protect the public’s right to free speech. But for speech to be truly free, we must also have the freedom to see or hear things that some people may consider objectionable,” he added.
It comes after Mr Musk endorsed a tweet saying that Jewish people harbour "hatred against whites".
Mr Musk was responding to the tweet, which read: "Jewish communities have been pushing the exact kind of dialectic hatred against whites that they claim to want people to stop using against them. I'm deeply disinterested in giving the tiniest s--- now about western Jewish populations coming to the disturbing realisation that those hordes of minorities that [they] support flooding their country don't exactly like themtoo much."
Mr Musk, who has 163 million followers, responded: "You have said the actual truth."