"There's an expression, but under the rules of #MeToo, I am not allowed to use that expression anymore". "It took a great man to get me to the make the switch", Cohen said at the time on Twitter, referring to Trump.
The contract, which was signed days before the 2016 presidential poll, also led to the violation of Trump campaign's finance laws, which Cohen had agreed later.
"I would do it except for these people [media] up there", Trump said when a rally attendee asked him to use it anyway.
"So, there is an expression, but we'll change the expression. That's pretty good", he said amidst applause from the audience.
Pope Francis praises resigning US Cardinal amid sexual abuse cover-up allegations
The Pope said Friday that he had also received a September 21 request from Cardinal Wuerl that his resignation be accepted. He was ordained a priest for the Diocese of Pittsburgh in 1966 and went on to receive a doctorate in 1974.
Michael Cohen, the president's former attorney and fixer, is now a registered Democrat, marking his latest move breaking away from the president since being charged by federal authorities.
At the rally, the president took aim at Democrats over their treatment of Supreme Court Justice Brett Kavanaugh, and lodged his most direct allegation yet against Hillary Clinton.
WATCH: Trump defends holding rally as Hurricane Michael devastates southeastern U.S. While mocking Senator Elizabeth Warren's heritage, Trump said he would have to "gently" toss a DNA test Warren's way so as not to hurt her "because we are in the MeToo generation".
The President denies all the allegations against him.