FOX Business' Charlie Gasparino is reporting that the billionaire investor is bidding $2 billion for the franchise and another $2 billion for the team's SNY cable broadcast home.
Cohen has been the name mentioned most as being interested in the club.
Earlier this year, the Wilpon family was close to selling most of the Mets franchise to Cohen for $2.6 billion, but Cohen walked away from the deal over control issues.
Interest in the team has increased recently with Vincent Viola, the former head of the New York Mercantile Exchange, and owner of the Florida Panthers hockey team, mentioned as looking into joining an investor group led by former Yankee Alex Rodriguez to buy the team.
Thursday was the deadline for the first round of bidding for the franchise.