The Italian striker moved to Anfield from Roma in July 2012, but scored just twice in 20 appearances as his season was hampered by a broken foot and dislocated shoulder.
Borini and Dossena are Sunderland's 12th and 13th signings of the window with the former the second frontline striker to link up with Paolo Di Canio's side after the arrival of United States international Jozy Altidore.
Dossena has signed a one-year deal at the Stadium of Light and also has previous Premier League experience at Liverpool.
The left-back made 26 appearances in all competitions last season but could not do enough to convince Rafael Benitez, the man who signed him for Liverpool, of his first-team qualities at Napoli.
Sunderland have begun their season in shaky form, failing to win a game from their opening three fixtures and picking up just one point, at Southampton and sit 19th.
Di Canio will also have to make do without Stephane Sessegnon, with the Benin international completing a move to West Brom.
