Giuliani spars with 'proven liar' Avenatti, who calls him 'train wreck of a lawyer' amid Cohen tape release

Trump lawyer Rudy Giuliani and Stormy Daniels attorney Michael Avenatti escalated their war of words on Thursday, amid reports of a secret audio recording showing that Trump's former attorney, Michael Cohen, had arranged to buy Daniels' silence on his own.

Trump later reimbursed Cohen for that $130,000 payment, according to Giuliani. Daniels, an adult film star whose real name is Stephanie Clifford, sued Trump in March, claiming the nondisclosure agreement she signed concerning an alleged Trump liaison wasn't valid because it lacked a Trump signature.

She later sued Trump for defamation, saying he'd unfairly hurt her reputation by dismissing as a "con job" her account of being threatened into silence about her story in 2011.

"Please be warned that Avenati [sic] is becoming desperate since he is being ignored," Giuliani tweeted Thursday morning. "The  Cohen-Cuomo tape makes clear Daniel claim is dead and with it Avenati’s meal ticket. Most media now consider him too much of a proven liar to put on. Any reliance is gross negligence."

In the purported conversation, Cohen spoke with CNN's Chris Cuomo and acknowledged on tape that he'd arranged — by himself — payment to Daniels in 2016. Cohen reportedly assured Cuomo that he was not taping their conversation, and put the phone in his desk drawer. The phone appears to record the whole conversation.

Avenatti fired back at Giuliani, who is also New York City's ex-mayor, almost immediately, calling him a "train wreck of a lawyer."

"Each time you open your mouth, it is a disaster," went Avenatti's retort. "Where are all the tapes? Release them! Our case is over because Trump’s criminal co-conspirator said so? Let’s debate the facts and the law -- any show any time. Put up or shut up!"

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He added: "@RudyGiuliani Call your good buddy @seanhannity and let's go on his show and discuss the facts and the law relating to Stormy Daniels, Michael Cohen and Mr. Trump. Let's do it any day next week. And if you will not do this, you need to pipe down and go crawl back under that rock."

Shortly after midnight on Thursday, Avenatti made additional claims.

"Breaking: Donald Trump conspired with Michael Cohen to pay off multiple other women prior to Election Day in 2016," he wrote. "They were also concerned about a pregnancy. Cohen has evidence and info in his possession and it must be released to the public."

Earlier this week, CNN aired a taped conversation in which Cohen and Trump appear to discuss the possibility of buying the rights to a Playboy model’s story about an alleged affair with Trump -- as with Daniels, the allegations involve a time period before Trump was president.

The release of that tape, which cuts off abruptly, prompted backlash from Trump and his allies.

“What kind of a lawyer would tape a client? So sad! Is this a first, never heard of it before? Why was the tape so abruptly terminated (cut) while I was presumably saying positive things? I hear there are other clients and many reporters that are taped - can this be so? Too bad!” Trump tweeted.

Fox News' Lukas Mikelionis and Brooke Singman contributed to this report.

Gregg Re is an editor for Fox News. Follow him on Twitter @gregg_re.