Mr Avenatti posted on Twitter a statement by Daniels that said she apologised to her fans in Columbus, but she would not perform her previously scheduled Thursday night show.
"It was an absurd use of law enforcement resources", Avenatti said.
Avenatti said the arrest was a politically-motivated "sting operation".
The president had claimed that the payment to Clifford - who says she had an affair with Trump - had nothing to do with his election campaign.
An arrest affidavit said Daniels allegedly inappropriately touched male and female undercover police officers after they threw cash in her direction.
Columbus police charged Daniels, whose real name is Stephanie Clifford, with three misdemeanor counts of touching a patron at a "sexually oriented" business in violation of an OH strip club law, according to online court records.
Daniels, whose real name is Stephanie Clifford, is in the middle of a scandal that some say could topple the US president.
Avenatti expected her to be released on bail shortly and vowed to "vehemently contest all charges", he tweeted.
OH strip club law reportedly prohibits patrons from touching a nude or seminude dancer unless the patron is a member of the dancer's immediate family. He says they will fight all charges, which he called "bogus".
Representatives for the Columbus Division of Police did not immediately respond to ABC News' request for comment. "There has to be higher priorities!", he added. A person who answered the phone at the club declined to comment. Avenatti tweeted that his client would not perform at the club Thursday night.
She is trying to free herself from a non-disclosure agreement signed before the election, and suing over a "defamatory" tweet by the USA president earlier this year.
Through White House officials, Mr Trump has denied the affair, although Mr Cohen has admitted paying Daniels as part of the agreement.