Blues boss Frank Lampard is a huge admirer of the Scottish midfielder, who swapped Glasgow giants Rangers for the Premier League side in 2017.
Gilmour made 11 appearances across all competitions for Chelsea’s first team last term, but missed the end of the season due to a knee injury.
The 19-year-old required surgery on his knee and is yet to feature for the Blues in the current campaign, but is keen to start playing senior football again.
With Gilmour eager to regain sharpness and hopeful of making Scotland’s squad for next year’s Euro 2020, Chelsea could have a decision to make over his immediate future in January.
While Lampard is a fan of the midfielder and would like to keep him in the team, it remains to be seen if the Blues boss can offer him the playing time that he would like.
It is also unclear if Gilmour will push for a loan move should he not get the playing time that he wants when the transfer window opens in January.
However, a host of clubs are expected to be in line should Chelsea open up to the prospect of allowing Gilmour to leave on loan in the winter.