The 20-year-old managed just 12 appearances in all competitions for the Blues last season, and Thomas Tuchel is believed to be keen to give the Scottish international the chance to seek more game time next year.
Gilmour made his first start for Scotland against England last Friday and his impressive performance earned him UEFA's Star of the Match award.
However, since then the Chelsea midfielder has tested positive for coronavirus and will now have to isolate for 10 days, missing Steve Clarke's side's crucial final group game against Croatia on Tuesday.
The Sun reports that recently-promoted Norwich are among five other top-flight clubs interested in taking Gilmour on loan next season, but manager Daniel Farke will hope to use his relationship from working with Tuchel at Borussia Dortmund to persuade Chelsea that Carrow Road is the right place for the youngster.
Tottenham Hotspur midfielder Oliver Skipp excelled with the Canaries on his loan spell last season which is an example of how Farke can help young players develop.
After graduating from the Chelsea academy when Gilmour made his senior debut in 2019, he has gone on to make 22 appearances for the club. body check tags ::