The 30-year-old left the club in May following the expiry of his contract but Hodgson offered him the chance to train with the Reds in a bid to earn a return.
Having impressed the new boss and his backroom staff, he has duly been handed a two-year contract to bring him back to the club until 2012 and was included in the travelling party for tomorrow's friendly against Borussia Monchengladbach.
Hodgson told Liverpoolfc.tv: "I asked the club to find out whether Fabio had found a new club.
"When we ascertained that he hadn't, we invited him back to Melwood so myself and my staff and the medical team could properly assess the player.
"We were very impressed and I am delighted we were able to re-sign Fabio."
Aurelio made 71 appearances in four seasons, scoring three goals, after being brought to Anfield by former boss Rafael Benitez.