ABC News additionally reported that investigators have asked Cohen whether the president or his associates had discussed a possible pardon for the lawyer, who pleaded guilty last month to federal criminal charges.
In addition to Russian attacks on the election, and possible Trump campaign collusion, Mueller is known to be probing whether Trump has obstructed justice in the investigation. Trump has repeatedly denied collusion and obstruction.
Trump's lawyer, former New York Mayor Rudy Giuliani, told NBC that "Cohen has a history of lying" in response to ABC's report.
The special counsel's office declined to comment, while the White House did not immediately respond to a request for comment.
Lanny Davis, a lawyer for Cohen, declined to comment.
Later on Thursday, The New York Times confirmed ABC's reporting that Cohen had been interviewed multiple times by Mueller's prosecutors, citing two people with knowledge of those interviews.
Vanity Fair reported last week that Cohen has in recent weeks been talking to investigators in Mueller's office.
The new article by ABC adds details about that reported contact. Among the bylines on the new report is ABC News anchorman George Stephanopoulos, who in July scored the first in-depth interview with Cohen after the lawyer's offices and residences were raided by FBI agents in April.
ABC News said Cohen has spoken to Mueller's team in both Washington and in New York City. The article said he is also cooperating with New York state authorities in an ongoing investigation into Trump's charity and the Trump Organization.
Cohen pleaded guilty to charges brought by federal prosecutors in New York City without having an agreement to cooperate with them or with Mueller.
However, his voluntary cooperation could be used by his lawyers, and prosecutors, to argue to a judge that he deserves leniency when sentenced for tax crimes, and for violating campaign finance laws by paying porn star Stormy Daniels $130,000 right before the election to keep quiet about an alleged affair with Trump. The president has denied that affair, but did reimburse Cohen for the payoff.
Read the full ABC News report here.