The Serie A outfit made a surprise £8m offer for the 23-year-old as former Chelsea and Tottenham Hotspur head coach Antonio Conte looks to bolster his midfield options.
The bid was rejected by the Seagulls, who are now bracing themselves for an increased offer expected to be in the region of £12m.
Romano tweeted: “Napoli are preparing new formal bid to send to Brighton for Billy Gilmour.
“Gilmour, open to joining Napoli. #BHAFC already rejected opening official bid from Napoli last week, revealed by @MatteMoretto.
“Napoli will return for Gilmour and also plan to sign [Marco] Brescianini [from Italian club Frosinone].”
Gilmour signed for Albion from Chelsea for £7.5m in 2022 and has two years remaining on his contract.
The Scotland international is currently in Japan with the Seagulls as part of a two-match tour.
