Mr Davis told MSNBC News: “I can’t tell you the answer about contacts between Michael and the Special Counsel.
“But I can tell you that Mr Cohen has knowledge on certain subjects that should be of interest to the Special Counsel and is more than happy to tell all that he knows.
“Not just the obvious possibility of a conspiracy to collude and corrupt the American democratic system in the 2016 election.
“But also knowledge about the computer crime of hacking and whether or not Trump knew ahead of time about that crime.”
Michael Cohen was Trump’s lawyer between 2006 and May 2018.
Special Counsel Robert Mueller is investigating alleged Russian interference in the 2016 presidential election, and whether there was any collusion with the Trump campaign.
Trump has repeatedly described the inquiry as a “witch hunt”.
On August 1 the US leader tweeted: “Russian collusion with the Trump Campaign, one of the most successful in history, is a total hoax.
“This is a terrible situation and Attorney General Jess Sessions should stop this rigged witch hunt right now.”
Mr Cohen pleaded guilty to violating campaign finance laws at a Manhattan court on Tuesday.
He claimed this was “in coordination and at the direction of a candidate for federal office”, implying President Trump.
Mr Davis added on Twitter: “Michael Cohen took this step today so that his family can move on to the next chapter.
“This is Michael fulfilling his promise made on July 2nd to put his family and country first and tell the truth about Donald Trump.
“Today he stood up and testified under oath that Donald Trump directed him to commit a crime by making payments to two women for the principle purpose of influencing an election.
“If those payments were a crime for Michael Cohen, then why wouldn’t they be a crime for Donald Trump?”
Mr Cohen’s sentencing will take place on 12 December.
The White House is currently declining to comment on Mr Cohen’s plea.