The president has turned on Mr Cohen, who was his personal lawyer and fixer for approximately a decade, and accused him of lying.
McDougal, a former Playboy playmate, and Clifford, an adult-film star, are two women for whom Cohen has said he arranged hush payments at Trump's direction to hide alleged affairs with Trump.
"His lawyers said that they went to my lawyers and asked for pardons", Trump said. "He authorized me after July 2 to say he would never accept a pardon if offered and I did".
Cohen also testified to Congress that he spoke directly to Mr Sekulow about pardons, according to the sources, which Mr Sekulow denies.
It came as Cohen, 52, prepares this spring to begin a three-year prison term, following his guilty plea to campaign finance violations. He's due to report to prison in May. He was also a Trump campaign advisor. He said Cohen "directed his attorney" to explore a pardon.
Cohen's lawsuit also renews an allegation that's dogged Trump for much of his career: He has a penchant for cheating people out of money he had promised them.
Kelly Back Behind Bars As His Money Problems Gets Worst
The 52-year-old singer is accused in Chicago of sexually abusing four females dating back to 1998, including three underage girls. The arrest comes less than two weeks after Kelly put up $100,000 to be released from custody on criminal sexual abuse charges.
Latest appointee to President Donald Trump's legal team and former Mayor of New York City Rudy Giuliani attends the Conference on Iran on May 5, 2018, in Washington, D.C. Giuliani, who said "truth isn't truth", asked if Democrats "care about the truth".
Michael Cohen testified last Wednesday before the House Oversight Committee in an open hearing.
"Over the last several years, President Trump has evaded accountability for his near-daily attacks on our basic legal, ethical, and constitutional rules and norms", House Judiciary Committee chairman Jerry Nadler said.
As a result, Cohen hired lawyers, and the Trump Organization, then run by Trump's sons, Eric and Donald Jr., promised to pay them, the lawsuit said.
Special counsel Robert Mueller is looking into whether the Trump campaign coordinated with Russian Federation and whether the president obstructed the investigation. "Lanny Davis coordinated and choreographed this whole thing", Jordan said.
Separately on Thursday, Cohen filed a lawsuit in New York City claiming the Trump Organization broke a promise to pay his legal bills and owes at least $1.9 million to cover the cost of his defense.
In correspondence obtained by Fox News, Turner inquired about who Cohen was in contact with, specifically asking about whether that included, "Democratic Members or Democratic staff of [Senate Select Committee on Intelligence], [House Committee on Oversight and Reform] or [House Permanent Select Committee on Intelligence] prior to his appearances before House and Senate committees last week".