The Scottish midfielder swapped Glasgow giants Rangers from Chelsea in 2017 and is a highly rated talent at Stamford Bridge.
He notched up eleven appearances for the Chelsea first team last season, being lauded for the maturity of his game, before he missed the rest of the campaign due to a knee injury.
The 19-year-old required surgery on his knee and is yet to make a single appearance for Chelsea during the ongoing campaign.
Chelsea are soon expected to decide whether to loan Gilmour out in January, and according to Italian daily Il Corriere del Mezzogiorno, Napoli are interested in the young midfielder.
The Serie A giants have been keeping tabs on Gilmour and are considering taking him to Italy on a loan deal from Chelsea.
Gilmour wants to play more football to regain match sharpness as he wants to be in Scotland’s squad for next summer’s European Championship.
If the midfielder does not receive assurances over more game time in the second half of the season then he could be interested in a loan move in January.
It remains to be seen whether Chelsea and Gilmour would be prepared to agree to a move to Italy next month.