The 20-year-old is regarded as one of the Blues’ best young players, but he has yet to cement a regular role with the senior squad due to stiff competition.
He is likely to be loaned out for the 2021/22 campaign, and it is claimed that the Canaries and Wolves could fight it out to land his signature.
The Canaries are optimistic about winning the race with manager Daniel Farke having a close relationship with Blues manager Thomas Tuchel.
Farke had previously coached the reverse side at Borussia Dortmund when Tuchel was the first-team head coach.
The Scotland international has made a big impression in his short career with his excellent distribution and ability to control play from the centre of the park.
He has still struggled for game time with the fierce competition for places. He is behind N’Golo Kante, Jorginho and Mateo Kovacic in the pecking order.
The Blues have Ruben Loftus-Cheek returning after a loan stint, and it appears ideal for Gilmour to gain valuable Premier League experience on loan.
The Canaries had Oliver Skipp on loan from Tottenham Hotspur last term, who was a key performer for them in the defensive midfield position.
They are unlikely to retain his services, and Gilmour could be a perfect replacement.
However, they face competition from Wolves, who could free up a space for Gilmour with the potential exit of Ruben Neves this summer.
The Midlands outfit are an established outfit in the top-flight unlike Norwich, but the latter will hope to benefit from the Farke-Tuchel friendship.
Stats from Transfermarkt.com