The Italian business newspaper Il Sole 24 Ore said the decision was taken on Wednesday in a meeting at the Genoa headquarters of the ship's owners, Costa Crociere, which is owned by Carnival Corp.
The journey from the island of Giglio -- where the wreck lies -- and Genoa will take around five days.
A number of ports including one in Sunderland bid for the job to scrap the ship. According to sources, many of the other bids were cheaper, but their distance from Giglio and the fragility of the wreck ruled them out.
According to the reports, a number of observers will travel alongside the ship, including Greenpeace.
At time of going to press, Costa Crociere was unavailable for comment.
Costa Concordia hit a reef off the island of Giglio on January 13, 2012, killing 32 people. It was partially raised in a complex "parbuckling" operation last September and currently sits on a manmade platform as floats (or sponsons) are attached to its sides, ahead of the towing.
--By Adam Coulter, UK Editor