"If I'm not up to it, I don't know who is", he said, before adding that he knows how the Department of Justice works better than most. Cohen had previously said Trump did not reimburse him, and Trump had denied knowing about the payment.
Trump has openly praised Giuliani since the former New York City mayor began hitting the airwaves in defense of the President on Wednesday.
Giuliani said there was no truth to the idea that unless he is more careful on TV, the president will oust him.
Senior administration officials are complaining they are still in the dark about when and where Giuliani will pop up on their television screens, one senior White House official said.
Daniels, whose real name is Stephanie Clifford, sued Mr Trump in March to get out of a "hush agreement" over their alleged relationship in which she was paid US$130,000 (S$174,000) by Cohen to keep quiet.
Chicago Police Department weighs in on Bryant's tickling incident
Chen (1-2) got tagged for nine runs and seven hits over three innings in his third start back from a strained left elbow. The plus money with Miami is tempting, considering that The Cubs may be a bit overpriced given their recent run.
In terms of his legal skills, I think it's clear that Giuliani has lost a step.
He also managed to voice a confident-sounding, if wholly fictitious, rationale for Trump to stonewall special counsel Robert Mueller. Giuliani's aggressive defense of the president in recent weeks has pleased Trump, but exasperated White House aides and attorneys.
But Giuliani has yet to become completely ostracized by Trump.
Giuliani suggested during an interview on "Fox & Friends" on Thursday morning that such an allegation being made public would have been politically damaging to Trump.
After his interview with Hannity, Giuliani said his revelation about the reimbursement was made with Trump's approval.
The president has not publicly discussed dismissing Giuliani and has appreciated his sharp attacks on the Russian Federation investigation and his forceful battles with the press, according to three White House aides and outside allies. "It was intended as a personal, embarrassing, harassing claim".