Gilmour is down the pecking order at Stamford Bridge and could face a season of limited game time if he stays at Chelsea.
Chelsea are prepared to let Gilmour go to Everton as part of a deal to land Toffees winger Anthony Gordon.
Frank Lampard is a fan of the Scotland international, but Brighton are plotting to beat Everton to Gilmour’s signature and take him to the south coast.
Brighton believe Chelsea would accept a bid in the region of £10m for Gilmour.
The Seagulls recently sold Marc Cucurella to Chelsea and are now looking to raid Stamford Bridge to take a player in the other direction.
Gilmour spent last season on loan at Norwich City, but was unable to help the Canaries avoid relegation to the Championship.
Brighton are firm admirers of the 21-year-old and want to see him continue his development under the watchful eye of Graham Potter.