The performer was arrested last week but charges were dropped.
The adult film star is bringing her act to Donald Trump's doorstep as he announces his pick for Supreme Court Justice.
Avenatti said he ultimately believes Cohen will “flip” on the president if prosecutors offer him a deal in return for cooperation.
Avenatti said a possible gag order should go “both ways by extending to all parties,” including Trump and Giuliani, who have both been vocal about the case on TV and Twitter.
Prosecutors were to meet with Daniels to discuss their investigation into Trump's former fixer Michael Cohen.
“I am headed down in a week. Don't worry,” she tweeted in response to someone asking her to help the migrant kids.
“What’s going to happen here is the narcissism of this president, his arrogance, his stupidity, is ultimately going to catch up to him," Avenatti said of Trump.
“Rest assured, I’m here at the @WhiteHouse w my family, feeling great, & working hard on behalf of children & the American people!” Melania Trump tweeted during her absence.
Michael Avenatti has frequently attacked President Donald Trump's personal lawyer, Rudy Giuliani.
“If we weren’t a threat, none of this would be happening,” Avenatti said.
“Absolutely, I have a lot of sympathy for the first lady,” Avenatti said.
“They have such a distorted image of who he is that they don’t have the kind of reaction that we do,” a Trump family author said of Ivanka, Melania and Donald Trump.
“I don't believe Mrs. Trump has ever discussed her thoughts on anything with Mr. Giuliani,” Melania Trump’s spokeswoman said.
“If any atty for any Fortune 500 co. made the public comments that Giuliani did yesterday (which he affirmed this morning), they would be immediately fired," Avenatti tweeted.
“You misogynistic fool," Mika Brzezinski said. "Are you kidding me? 'Just look at Stormy Daniels?' Just look at yourself. Are you kidding me?"
"There are numerous expenses associated with cases like this and here, security costs alone are astronomical," Avenatti tweeted.
“I think Donald Trump promised Melania that he would not win, [that] she didn’t have to worry about ever living in the White House,” Boehner said.
Avenatti’s claim came the same day a report alleged he personally contacted top Democratic operatives for financial help in his legal fight against Trump.
Avenatti also said he would be demanding access to any audiotapes featuring Stormy Daniels's former lawyer and Cohen.