The New York Times reported that Rosenstein signed off on the raid, in which the FBI seized emails, tax documents and records, some of them related to Cohen's payment to adult-film star Stormy Daniels.
FBI officials raided Cohen's office in Manhattan, along with his residence and a hotel room.
Cohen's lawyer, Stephen Ryan, said in a statement Monday that federal prosecutors in Manhattan obtained a search warrant after receiving a referral "in part" from special counsel Robert Mueller. Ryan called the search "completely inappropriate and unnecessary."
The raid enraged Trump, who on Tuesday condemned the move as a "disgrace" and a "pure and simple witch hunt."
“It’s an attack on our country in a true sense. It’s an attack on what we all stand for,” Trump said at the White House.
The fiery comments immediately sparked speculation that the FBI raid on Cohen’s office could lead Trump to take steps to try and fire Mueller, who is investigating Russian election interference and possible ties between the Trump campaign and Russia.
“We’ll see what happens. … Many people have said ‘you should fire [Mueller],’” Trump said. “Again, they found nothing and in finding nothing, that’s a big statement.”
Trump's advisers are urging him to avoid making an impulsive decision in response to the raid, according to the Times report.
If Trump were trying to get rid of Mueller, he would need a sign off from Rosenstein, who appointed Mueller and oversees the special counsel's office because of Attorney General Jeff Sessions's decision to recuse himself.