Gilmour came through the ranks of Rangers academy and joined Chelsea in 2017.
Frank Lampard gave the midfielder his Chelsea debut against Sheffield United in 2019 after he came off the bench to replace Tammy Abraham.
Last season, he was on loan at Norwich City, where he appeared in 28 games and provided two assists.
Several clubs were interested in signing the 21-year-old this season, but Chelsea were not interested in letting go of the Scot for a permanent transfer.
However, Thomas Tuchel has decided to let the player go on a permanent transfer and Brighton have agreed a £10m deal with Chelsea for Gilmour’s signature.
And now Rangers are set to receive 15 per cent of the transfer for the midfielder due to a sell-on clause in his contract.
Gilmour made a total of 22 outings for Chelsea during his stay with them and failed to make an appearance for the Blues this season.
