"John was an incredibly enthusiastic defender who was always a great pro," he told the club's official website.
"He worked hard and never gave us a minute's trouble. The fans will remember that powerful shot of his that brought them to their feet on many occasions.
"We had a good chat about his time here and he wants to thank the fans for the support he was given down the years. He secured Champions League and FA Cup success at Anfield and also helped us win the Worthington Cup and two European Super Cups."
Dossena, who has been capped once by Italy, will become manager Rafael Benitez's second signing of the summer following the arrival of Swiss full-back Phillip Degen on a free transfer.