Daniels, whose legal name is Stephanie Clifford, filed the complaint in federal court in NY on Monday.
In the tweet, Trump dismissed her claims of being intimidated into signing a non-disclosure agreement by an anonymous man when he said, "A sketch years later about a nonexistent man. A total con job, playing the Fake News Media for Fools (but they know it!)" Trump wrote. "Mr. Trump used his national and global audience of millions of people to make a false factual statement to denigrate and attack Ms. Clifford".
Mr. Trump's representatives have denied that he had a sexual encounter with Ms. Clifford.
Mr Trump's lawyers did not immediately respond to requests for comment.
Daniels, whose real name is Stephanie Clifford, detailed her account of their affair on 60 Minutes last month.More news: PSG an elite team thanks to Emery, says club president
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Avenatti suggested on Twitter Monday that Trump knew about the alleged threat and had some role in it.
Daniels first told CNN in March about the alleged 2011 threat, which came soon after Daniels attempted to sell her story about the affair to InTouch magazine. We fully intend on bringing it to light.
The sketch depicted a man Daniels described as having threatened her in a Las Vegas parking lot in 2011.
Trump's "false statement about Ms. Clifford is defamation per se because, among other things, it charged Ms. Clifford with committing a serious crime", the lawsuit claimed. Cohen has admitted to paying Clifford $130,000 for her silence just days before the 2016 election. Thus, Mr. Trump may have actual knowledge of the incident and of the falsity of his statement.
Trump is appealing a March 20 decision by a NY state judge in Manhattan that allowed Summer Zervos, a contestant on his former TV show "The Apprentice", to pursue her own defamation case. That civil lawsuit was delayed in federal court in Los Angeles on Friday, with the judge citing a criminal investigation that Cohen is facing. Is the new lawsuit an attempt to keep Daniels in the limelight while the other case is on hiatus?