The Serie A outfit have been pushing to sign the Scotland international throughout this summer since the arrival of new head coach Antonio Conte.
It was reportedly last month that Napoli failed with an £8m offer to sign Gilmour, but were made aware that Brighton would be willing to sell the midfielder if they returned with an acceptable bid.
Gilmour came on as a second-half substitute in Brighton's 3-0 opening win over Everton in the Premier League last weekend, and Seagulls supporters chanted "we want you to stay" to the midfielder from the away end.
Brighton chief executive and deputy chairman Paul Barber has previously stated that Gilmour is an "important" player and the club want to keep him, while new head coach Fabian Hurzeler recently suggested that he fully expects the Scotsman to stay this summer.
Asked by Sussex World earlier this week if Gilmour will be available to face Manchester United in the Premier League on Saturday and if he expects the midfielder to remain a Seagulls player after the transfer window closes, Hurzeler simply replied: "Yes and yes".
Gilmour out, O'Riley in at Brighton?
However, according to journalist Fabrizio Romano, Napoli have now reached a verbal agreement with Brighton to sign Gilmour for a fee between €17m (£14.4m) and €18m (£15.3m) including add-ons.
Romano adds that Napoli have initiated plans for the 23-year-old to travel to Italy and undergo medical tests early next week.
Gilmour briefly worked with new Napoli boss Conte during his time at Chelsea, but the Italian left Stamford Bridge shortly before the midfielder was handed his senior debut under Frank Lampard.
He went on to make just 22 senior appearances for Chelsea before joining Brighton in the summer of 2022, and he played 41 times for the Seagulls across all competitions last season, including 30 outings in the Premier League and eight in the Europa League.
A move to Italy now appears on the cards for Gilmour, though, and his departure is set to see Brighton step up their pursuit of Celtic star Matt O'Riley.
As reported by Romano, Brighton are advancing in talks to sign O'Riley and hope to finalise an agreement with Celtic soon, which could see the 23-year-old midfielder sold for a Scottish-record transfer fee in excess of £25m.