The 23-year old Scotland international, who has two years left on contract, is wanted by Antonio Conte and further discussions are expected this week with the Italians confident of reaching agreement.
Manager Conte worked with Gilmour at Chelsea, which is where the Scotsman made his breakthrough.
The midfielder has impressed at Brighton so far, where he made 32 appearances in the league last season.
Conte also wants to sign Romelu Lukaku again having previously enjoyed success with the striker at Inter Milan.
The 31-year old has a £38m release clause but Napoli are looking to offer around £25m and are waiting for talks to progress with Paris St Germain over the sale of Victor Osimhen.
Conte has so far signed Alessandro Buongiorno from Torino, Rafa Marín from Real Madrid and Leonardo Spinazzola from Roma for a combined total of £40m.
Brighton, meanwhile, are close to signing another midfielder in 21-year old Paraguay international Diego Gomez from Inter Miami for around £13.9m.
Author: Mujuni Henry
JUNIITV
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