Cohen last week gave dramatic testimony to the US Congressional House Oversight Committee about the president, during which he accused him of being a "racist", "conman" and a "cheat".
Davis, who was not Cohen's lawyer at the time, said Cohen "directed his attorney" to explore a possible pardon with Trump lawyer Rudy Giuliani and others on Trump's legal team.
Cohen arranged payments to Clifford, who goes by the stage name Stormy Daniels, and McDougal to prevent them from speaking publicly about alleged affairs with Trump.
As he traveled on Air Force One to tornado-ravaged Alabama, Trump lashed out at his former attorney, calling him a "bad lawyer" and a "fraudster" even though he kept him on his payroll for a decade.
Cohen said the organization has refused to pay any of his legal fees or costs since May 2018, and owed him about $1.9 million as of January 25, 2019.
Earlier this year, Cohen hired two new Chicago lawyers and parted ways with the attorneys who represented him for months as he cooperated with Mueller and prosecutors in the Southern District of NY.
"The fact that more independents are starting to question Trump's fitness for office could be a sign of trouble, if not for impeachment itself than for his re-election prospects in 2020", said Patrick Murray, director of the independent Monmouth University Polling Institute.
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The Wall Street Journal reported this week, however, that Cohen had asked his attorneys to pursue one. "I said NO. He lied again!" The results were heavily skewed on party lines, with most Republicans believing Trump and most Democrats trusting Cohen.
"The president is not going to issue pardons in this investigation", Giuliani said.
Cohen ultimately pleaded guilty and is co-operating against the president in separate investigations by the special counsel and by federal prosecutors in NY. He has pleaded guilty to lying to Congress because the actual end date was June 2016.
In a statement, a lawyer for the Trump Organization, Marc Mukasey, disputed Cohen's claims.
Cohen has become a key figure in congressional investigations after turning on his former boss and co-operating with special counsel Robert Mueller's probe. That continues to be the case.
"His statement at the Oversight Hearing was true - and consistent with his post-joint defense agreement commitment to tell the truth".
The lawsuit is likely to prolong the bitter public stand off between the two men, as Cohen heads to prison in May after pleading guilty to campaign finance violations, tax evasion and lying to Congress in August 2018 in a case brought against him by federal prosecutors in NY.
Cohen came under renewed scrutiny after telling Congress that he never sought a pardon from the president. Trump Jr. told the Senate Judiciary Committee in 2017 he was only "peripherally aware" of the proposal.