The former Liverpool defender came close to joining Torino on Friday, but failed a medical after agreeing personal terms with the Turin side.
Dossena has a thigh problem which Torino decided was enough to pull the plug on a proposed deal, instead moving to sign Giovanni Pasquale on loan from Udinese.
The breakdown of that deal has alerted Sunderland manager Paolo Di Canio, who is desperate to strengthen his squad before the transfer window closes tomorrow night.
Napoli want to offload Dossena and have entered negotiations with the Mackems over a season-long loan switch, according to Sky Sport Italia.
The 31-year-old has previous experience of playing in the Premier League following a two-year spell at Anfield in which he made 31 appearances.