Gilmour starred during Brighton’s 2-1 win over McTominay’s Manchester United on Saturday afternoon and new head coach Fabian Hurzeler is an admirer of the Scot.
But with two years left on his contract, Brighton are open to allowing the 23-year-old to depart the Amex Stadium ahead of the transfer window closing.
Antonio Conte has identified Gilmour and McTominay as key signings in a new-look Napoli midfield, with Chelsea striker Romelu Lukaku also set for a £30m move to Italy.
Napoli agreed a £25m package for McTominay over the weekend and want both Scotland internationals in Conte’s squad to face Parma in Serie A on Saturday.
Gilmour is waiting for the green light to travel and rubber stamp the switch after making 46 Premier League appearances for the Seagulls across two years.
Conte’s side lost their opening league game of the campaign 3-0 away at Verona before beating Bologna 3-0.
Both were without Victor Osimhen, who is looking for an exit