Fulham boss Martin Jol had been hoping to reach an agreement for the vastly-experienced Greece international, with an offer put on the table.
Torosidis has decided to snub the Cottagers' advances and make a move to Italy instead.
He was in Rome to undergo a medical on Tuesday and appears to have shown his new employers that he is in full working order.
The 27-year-old has now completed all of the formalities required to complete a transfer and is all set to continue his career in the Italian top-flight.
"Olympiakos can announce that an agreement has been reached with AS Roma for the transfer of Vasilis Torosidi," read a statement on the Greek club's official website.