Chelsea are understood to have agreed a fee for around £9million with the Seagulls for the midfielder.
The 21-year-old has not had much of a look-in at Stamford Bridge under Thomas Tuchel. He was loaned out to Norwich City last season and it was expected he would depart having not been given a squad number for the campaign.
Gilmour was eventually given No.35 and was an unused sub during the 2-1 defeat to Southampton on Tuesday. He will not continue his football career away from Chelsea where he made 22 first-team appearances, the majority under Frank Lampard.
According to The Times, the player will sign a four-year-deal with an option for another.
With 15 caps for Scotland, Gilmour, who left Rangers for Chelsea in 2017, is a key component of Steve Clarke’s side. Regular football in the Premier League under the tutelage of Graham Potter will be beneficial for player and country.
He will become Brighton’s fifth summer signing.
Meanwhile, Stephen Kelly is on the verge of signing a permanent deal with Livingston.
The Scottish Sun reports Livi have won the race for the 22-year-old Rangers midfielder.
Kelly is in the final year of his deal at Ibrox but didn’t feature in Giovanni van Bronckhorst's plans.
He has had three loan spells during his time at Rangers; at Ayr United, Ross County and most recently at Salford City.
The player recently spoke about his future, noting he is fit to hit the ground running wherever he moves.