The 31-year-old midfielder is expected to leave Sunderland this summer.
The 31-year-old will return to the Black Cats for pre season after spending the 2015-16 campaign on loan with Bologna, impressing after scoring seven times in 27 Serie A starts.
The Italy international joined the Italian side while Dick Advocaat was Sunderland boss and it remains to be seen whether current manager Sam Allardyce - who replaced the Dutchman early in the 2015-16 season - rates him.
The Echo claims that Giaccherini's stock is "extremely high" in Italy, with Torino and Fiorentina reportedly interested, but the report goes on to claim that Rome-based outfit Lazio have joined the race for his signature.
It may be in Sunderland's interest to hold off on a sale until after Euro 2016. If Giaccherini impresses, they stand a much better shot over at least recouping the £6 million that then-Black Cats boss Paolo Di Canio paid for him back in 2013.
The opposite is also true. A dismal showing may see his value decrease. Is it worth the risk from a Sunderland perspective?