The statement came hours after Cohen pleaded guilty to campaign finance violations and other federal crimes.
Cohen's plea came at the same time a jury in Virginia convicted former Trump campaign chairman Paul Manafort on eight counts of fraud and other charges. The president, meanwhile, was headed to West Virginia for a campaign rally on what has become one of the most consequential and perhaps catastrophic days of his presidency.
Cohen did not name Trump himself Tuesday, when he admitted in court to various crimes, and prosecutors did not mention the president's name in the plea agreement.
However, Cohen said "a candidate" for federal office directed him to make illicit payments to two women meant to influence the 2016 elections.
"If those payments were a crime for Michael Cohen, then why wouldn't they be a crime for Donald Trump?" Davis said in a statement. In addition to being Cohen's lawyer, Davis served in the White House under President Bill Clinton.
Trump has denied involvement with either women, porn star Stormy Daniels and Playboy model Karen McDougal.
Rudy Giuliani, Trump's lead outside lawyer in the special counsel's Russia probe, meanwhile, slammed Cohen.
"There is no allegation of any wrongdoing against the President in the government's charges against Mr. Cohen," Giuliani said in a statement. "It is clear that, as the prosecutor noted, Mr. Cohen's actions reflect a pattern of lies and dishonesty over a significant period of time."
Trump ignored questions about Cohen when he landed in West Virginia, although he weighed in on the Manafort conviction.
"Paul Manafort is a good man. ... It doesn't involve me but it's a very sad thing. ... It had nothing to do with Russian collusion," Trump said.
The president has repeatedly denied collusion or obstruction of justice.