The Sun says the England winger is in talks about a move to Italy with the Serie A champions this week.
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Sterling is a free agent after terminating his contract with Chelsea last month.
The veteran would move to Napoli as cover for former Manchester City teammate Kevin de Bruyne, who is currently sidelined with injury.
Sterling, 31, is willing to accept a major drop on his wages to make the move happen.
Napoli’s sporting director Giovanni Manna confirmed last week they had been in contact with Sterling's camp, though his wage demands were proving an issue.