He called on Trump and his former longtime personal lawyer, Michael Cohen, to come forward and disclose information about the women and their supposed relationships with Trump.
"They should release the information to the American people now," Avenatti said. "Enough with the games."
Trump's attorney, Charles Harder, did not immediately respond to an email seeking comment late Thursday.
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"Last time I checked they weren't just handing out checks to anyone whether they had a relationship or not," Avenatti said.
Daniels, whose real name is Stephanie Clifford, has said she had sex with Trump in 2006, when he was married, but Trump has denied it. Days before the 2016 presidential election, she was paid $130,000 to stay silent in a deal handled by Cohen. She is suing to invalidate the nondisclosure agreement.
Cohen's attorney, Lanny Davis, would not say if he knew whether Cohen had been involved in any other hush-money deals on behalf of Trump.
