During the interview, Guthrie suggested that Avenatti's refusal to release evidence proving that Daniels' alleged 2006 affair with US President Donald Trump actually happened made it seem like he was "playing games" by sending out cryptic messages on Twitter.
"She's answered some very difficult questions… we're not going to play our hand right now and the reason is because we have a case to win," Avenatti told Guthrie moments before the journalist brought up his Thursday tweet.
"I'm not playing games," Avenatti said. "There will be more evidence. It's just not going to happen right now. Everyone wants immediate gratification. It's not going to happen right now."
Shifting to a different topic, Guthrie later brought up the portion of the "60 Minutes" interview when Daniels recounted an unidentified man threatening her to keep silent about the affair.
"There was a specific reference by that individual to Mr. Trump, so it's pretty clear that it could've only come from one place only," Avenatti said. "It had to be someone that was either related or sent by Mr. Trump or Mr. Cohen… someone from that side had to have directed it because it would have made no sense for it to come from any place else."
The Los Angeles-based lawyer went on to note that Daniels had shared more details about the Trump affair in the interview, but that footage wound up on the cutting room floor due to the show's time restraints.
"She was prepared to discuss intimate details relating to Mr. Trump," Avenatti told Guthrie. "She can describe his genitalia. She can describe various conversations that they had that leave no doubt as to whether this woman is telling the truth."
"And if she's not telling the truth, let the president take to the podium and call her a liar. Let the president come forward and say it never happened. There is a reason why this $130,000 was paid and it wasn't paid because she made this story up," he added.
However, while Avenatti continued to promise more forthcoming details, a lawyer representing Michael Cohen, Trump's personal lawyer, nonetheless sent a "cease and desist" letter to Daniels after the actress implied that Cohen was in some way linked to the incident, according to the Washington Times.
Aside from demanding that Daniels stops spreading "false and defamatory statements," the letter also requested that Avenatti and Daniels "immediately retract and apologize" for statements suggesting Cohen had anything to do with the threat.
During the White House' press briefing Monday, Principal Deputy Press Secretary Raj Shah told reporters that Trump has "consistently denied" the affair allegations and that her claims are not accurate.
"The president doesn't believe any of the claims that Ms. Daniels made last night in the [60 Minutes] interview are accurate," Shah told NBC's Kristen Welker, before adding that POTUS doesn't believe Daniels was threatened.
"He just doesn't believe that… there is nothing to corroborate her claims," Shah explained.
The official never confirmed whether or not Trump actually sat down to watch the interview.
Daniels has since responded to the "cease and desist" order by suing Michael Cohen for defamation. The suit, according to The Hill, zeros in on a comment Cohen previously made last month that stated: "Just because something isn't true doesn't mean that it can't cause you harm or damage."