The Italian international is wanted in Serie A after an impressive Euro 2016.
Italy's Emanuele Giaccherini scores against Belgium
The 31-year-old was thought to have agreed deal with fellow Serie A side Torino, after they made a 1.8 million euro (£1.5 million) offer, but Napoli have now entered the race and have made initial contact with the players agent.
Italy's Emanuele Giaccherini celebrates
The report claims that personal terms are all that has to be established now, with Napoli presumably willing to match Torino’s offer, yet it remains unclear which path the Italian international will opt for.
Napoli are the bigger draw and can offer Champions League football however he will have a much harder time nailing down a first-team spot at the Stadio San Paolo.
Sunderland's Emanuele Giaccherini
Giaccherini may therefore decide Torino represents the more attractive option but whichever club he chooses, a deal seems likely to be done soon.
The Black Cats midfielder has struggled to adapt to English football but enjoyed an impressive Euro 2016 campaign with the Azzurri, leading to the current battle for his signature.
Sunderland manager Sam Allardyce
Meanwhile, Sam Allardyce has revealed his frustrations at the lack of incomings at the club so far this summer after the failure to secure the services of former Newcastle left-back Davide Santon.