The 27-year-old had been tipped to join the Premier League outfit this month, but Roma have confirmed that a deal is now in place.
"Yes, we were about to sign the contract with Fulham but Roma have made a better offer at the last minute," Kazianis told Calciomercato.
A statement from the Italian club said: "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."
The Greek midfielder has spent his entire career in his home country.