Their longtime relationship has been under increased scrutiny since the Federal Bureau of Investigation raided Cohen's home, Manhattan office and hotel room in April.
Davis will be working with Guy Petrillo, the NY lawyer who was recently hired by Cohen to represent him in an ongoing criminal probe into his business activities in the Southern District of NY, a person familiar with the matter said.
Cohen is under federal investigation in NY in connection with his business dealings. President Trump has denied the relationship.
"Once I understand what charges might be filed against me, if any at all I will defer to my new counsel, Guy Petrillo, for guidance", he said.
"His mindset is of someone who is operating under the assumption that he is not getting that, though of course he doesn't know one way or the other", the second friend said.
During their meeting, Cohen asked Trump "when business began", to which Trump replied, "Hundreds of millions of years ago people were doing business and they were trading in rocks and stones".
The video then says that Cohen is back "like you've never seen him before".
"It was disgraceful", continues Trump.
Cohen has famously said he would take a bullet for Trump and he has fashioned himself as Trump's "fixer", willing to help handle situations quietly.
In late November, former national security adviser Michael Flynn notified Trump's legal team that he would end a similar agreement.
Shayanne Gal/Business Insider; Alex Wong/Getty; Jonathan Ernst/ReutersSpecial Counsel Robert Mueller is investigating the Trump election campaign's possible ties to Russian Federation.
Lanny Davis, Cohen's adviser, is declining to comment on any issue involving the investigation.