Daniels' newly filed lawsuit says that after the news broke, Cohen texted Davidson on Jan. 17, 2018, and told him "I have her tentatively scheduled for Hannity tonight," a reference to Fox News host Sean Hannity.
That message and others purportedly exchanged that day between the two men were sent on the same day that they became aware that In Touch magazine was publishing an old interview with Daniels in which she discussed the alleged affair.
The suit says that Cohen was desperate to have Daniels appear on Fox News to deny the affair, even after Davidson said she would not be able to appear on Hannity's show that particular evening.
"I'm trying to get her to commit for tomorrow," Davidson texted Cohen, the suit says.
The suit goes on to say that Cohen, "in a panic," sent several other texts "in a desperate attempt to convince Ms. Clifford, with the assistance of Mr. Hannity, to lie to the American public about her relationship with Mr. Trump via the Fox News broadcast."
According to the suit, in one of those subsequent texts, Cohen wrote to Davidson: "This is no good. We need her as by doing tomorrow you just create another news cycle instead of putting an end to this one."
But less than three hours later, Cohen allegedly texted Davidson again, saying, "Keith, The wise men all believe the story is dying and don't think it's smart for her to do any interviews. Let her do her thing but no interviews at all with anyone," according to the suit.
"Thanks pal," Cohen wrote, after allegedly getting Davidson's agreement, the suit says.
The suit says Cohen's reference to "wise men" included "Mr. Trump." It does not offer any proof for that claim.
Fox News did not immediately respond to a request for comment.