But Trump tweets: "Sorry, I don't see Michael doing that despite the disgusting Witch Hunt and the dishonest media!"
The FBI raided Mr Cohen's office, home and hotel room earlier this month.
President Donald Trump on Saturday said he doesn't expect his personal lawyer Michael Cohen to "flip" and cooperate with federal prosecutors in their investigation of Cohen's business dealings, including a payment to a porn actress to protect Trump.
The high-profile White House correspondent has had remarkable access to the president and his inner circle, which she's used to land some of the biggest palace intrigue stories of the Trump White House - including about Trump's recent "meltdown" following the Cohen raid.
"The guy who's really sweating it out these days is Michael Cohen, Trump's criminal lawyer-and when I say 'criminal lawyer, ' I mean lawyer who's a criminal", Maher joked during his monologue.
The Pulitzer was awarded jointly with the Washington Post for "deeply sourced, relentlessly reported coverage in the public interest that dramatically furthered the nation's understanding of Russian interference in the 2016 presidential election and its connections to the Trump campaign, the president-elect's transition team and his eventual administration".
Hannity had not previously disclosed his relationship with Cohen until it was revealed by the lawyer under pressure from a NY judge.
Trump also lashed out personally at one of the story's writers, calling reporter Maggie Haberman "third rate" and claiming he has "nothing to do with" her. Trump later deleted and reposted the tweets correcting the spelling of Haberman's name.
'Michael is a businessman for his own account/lawyer who I have always liked & respected, ' he continued. Most people will flip if the Government lets them out of trouble, even if. "The person replied, 'He's abusive to everybody'". He slammed Maggie Haberman, the lead reporter on a new story, and called a former aide quoted in the story as a "drunk/drugged up loser".
On Saturday morning, Trump defended Cohen in a flurry of tweets at around 8.20am which he deleted within an hour. The Times speculated that Cohen might not be as willing to sacrifice for Trump despite his previous statement that he would take a "bullet" for the president.
The lawyers said letting Daniels' case proceed could impair Cohen's constitutional right under the Fifth Amendment against self-incrimination.