According to The Athletic, Chelsea are keen to loan the midfielder out to a fellow Premier League club or another top-flight side in Europe.
Despite Gilmour's excellent performances for Scotland, he has not been part of Thomas Tuchel's plans at Stamford Bridge and has not even made a bench this season.
He spent the 2021/22 campaign with Norwich City, playing 28 times.
His best period at Chelsea in the first-team came under Frank Lampard. The last of his 22 appearances for the club was back in May 2021.
Meanwhile, another former Ibrox starlet Adedire Mebude is attracting interest from PSV Eindhoven, who Rangers beat in the Champions League play-off, and Club Brugge who are also in the group stage of UEFA’s elite club competition, as well as an unnamed Premier League club.
The 18-year-old winger left Rangers for Manchester City in 2020 and has impressed in the youth ranks of the Premier League champions.
Mebude, who has been capped at Under-19 level for Scotland, had an excellent campaign last season, hitting double figures in goals and assists as City won the Under-18s title.
The Athletic report that talks have been held regarding a permanent contract with a possible buy-back option. Although Manchester City have not ruled out extending the player’s contract.
The English transfer window closes on Thursday, September 1 at 11pm.