The 27-year-old had been tipped to leave Olympiakos and join fellow countryman Giorgos Karagounis at Craven Cottage.
However, a late move from Roma has seen them win the race for Torosidis, who has previously spent his entire career in his homeland.
"Yes, we were about to sign the contract with Fulham but Roma have made a better offer at the last minute," his agent George Kazianis to Calciomercato.
Roma have since confirmed that a deal is agreed with Olympiakos and the player is now due in the Italian capital to undergo a medical.
The club announced: "Vasilis Torosidis is speaking to Roma's directors after the club reached an agreement in principle with Olympiakos.
"The transfer is subject to the player agreeing contract terms and passing a medical."