Davis' latest comments cast doubt on what Cohen may know, including about a June 2016 meeting in New York's Trump Tower attended by Trump's eldest son and a Russian lawyer.
Trump and his allies seized on the erosion of those claims.
"Michael Cohen's attorney clarified the record, saying his client does not know if President Trump knew about the Trump Tower meeting (out of which came nothing!)," Trump tweeted. "The answer is that I did NOT know about the meeting. Just another phony story by the Fake News Media!"
Davis told the Washington Post that he cannot confirm media reports that Cohen is prepared to tell Special Counsel Robert Mueller that Trump had advance knowledge of the 2016 Trump Tower meeting, where Donald Trump jnr expected to receive damaging information about Clinton from a Russian lawyer.
CNN reported last month that Cohen was claiming to have witnessed Trump being informed of the Russians' offer by Trump jnr and that the then-candidate approved the meeting.
The following day, the Washington Post reported that Cohen had told associates that he witnessed an exchange in which Trump jnr told his father about an upcoming gathering in which he expected to get information about Clinton. The Post did not report that Trump jnr told his father that the information was coming from Russians.
The information in the Post story, which was attributed to one person familiar with discussions among Cohen's friends, came from Davis, who is now acknowledging his role on the record.
Davis said he should not have expressed such confidence in his information.
"I should have been more clear - including with you - that I could not independently confirm what happened," Davis said, adding: "I regret my error."
In the past week, when asked directly by CNN's Anderson Cooper whether there was information that Trump knew about his son's meeting with Russian lawyer Natalia Veselnitskaya beforehand, Davis said, "No, there's not."
In a statement, a CNN spokeswoman said, "We stand by our story, and are confident in our reporting of it."
People familiar with Cohen's testimony to the House and Senate Intelligence committees said Cohen was interviewed extensively about the Russian interference campaign but did not provide any information to suggest Trump had advance knowledge.
Senators Richard Burr, (R), and Mark Warner, (D), leaders of the Senate committee, said in recent days that Cohen had sent word to the committee that he had no desire to amend that testimony.
In his interview with the Post, Davis also hedged on an idea he widely promoted after Cohen's guilty plea: that the longtime Trump loyalist had information that Trump knew of the Russian hacking of Democratic emails ahead of time.
Davis floated that idea in numerous broadcast interviews in recent days, repeatedly touting his client's potential value to Mueller.
"I believe that Mr Cohen has direct knowledge that would be of interest to Mr Mueller that suggests - I'm not sure it proves - that Mr Trump was aware of Russian government agents hacking illegally, committing computer crimes, to the detriment of the candidate who he was running against, Hillary Clinton," Davis told PBS last week.
But asked at the weekend how confident he was that Trump knew about the hacking before it became public, Davis said: "I am not sure. There's a possibility that is the case. But I am not sure."
Davis said that in discussing the hacking allegations last week, he should have emphasised his lack of certainty. He said he raised the idea that Cohen might have information about Trump's knowledge because he had a strong feeling that might be the case.
"I was giving an instinct that he might have something to say of interest to the special counsel" about hacking, Davis said. In retrospect, he said, "I am just not sure."
