The Scotland midfielder swapped Glasgow giants Rangers for Chelsea in 2017 and is a highly rated talent at Stamford Bridge.
He made eleven appearances for Chelsea last season before sustaining a serious injury that forced him to miss the latter half of the campaign.
Gilmour made his first start for Chelsea this season at Krasnodar on Tuesday night and there are suggestions that the Blues could loan him out in the January transfer window, with the player keen on game time ahead of Euro 2020.
Rangers have been floated as a possibility for Gilmour, but it is claimed that Chelsea will not sanction a move to Ibrox.
Chelsea are insistent that they will not loan out the youngster to Rangers or to any other club in January.
Frank Lampard wants the Scottish midfielder to stay as part of his squad and Gilmour is a big part of Chelsea’s plans.
Serie A giants Napoli have also been considering signing the midfielder on loan, but it seems they are also unlikely to succeed.
Gilmour wants to play more football in order to be ready for the Scotland squad for next summer’s European Championship.
