President Trump said the fact checks were “editorial decisions” by Twitter and amounted to political activism.
He said it should cost those companies their protection from lawsuits for what is posted on their platforms.
Mr Trump's order directs executive branch agencies including the Federal Communications Commission and the Federal Trade Commission to study whether they can place new rules on the companies, though experts expressed doubts much could be done without an act of Congress.
It sets out to clarify the Communications Decency Act, a US law that offers online social platforms legal protection in certain situations.
Under Section 230 of the law, websites such as Facebook, Twitter and YouTube are not generally held responsible for content posted by their users, but they can remove content that is obscene, harassing or violent.
The executive order points out this legal immunity does not apply if a social network edits content posted by its users.
It also says "deceptive" blocking of posts, including removing a post for reasons other than those described in a website's terms of service, should not be offered immunity.
A similar executive order was previously considered by the administration but shelved over concerns it couldn’t pass legal muster and that it violated conservative principles on deregulation and free speech.
Two administration officials outlined the draft order on the condition of anonymity because it was still being finalised.
"This will be a big day for social media and fairness!" Mr Trump tweeted beforehand.
Mr Trump on Wednesday claimed tech giants "silence conservative voices".
He said: "We will strongly regulate, or close them down, before we can ever allow this to happen."
Mr Trump and his campaign reacted after Twitter added a warning phrase to two Trump tweets that called mail-in ballots "fraudulent" and predicted "mail boxes will be robbed".
Under the tweets, there is now a link reading "Get the facts about mail-in ballots" that guides users to a page with fact checks and news stories about Mr Trump’s unsubstantiated claims.
Mr Trump accused Twitter of interfering in the 2020 presidential election and declared "as president, I will not allow this to happen".
His campaign manager, Brad Parscale, said Twitter’s "clear political bias" had led the campaign to pull "all our advertising from Twitter months ago".
In fact, Twitter has banned political advertising since last November.