The 19-year-old, also known as Gabigol, has been targeted by Serie A champions Juventus, but the London club are keen to win the race by offering Santos £25million for the striker.
The Brazilian club are hopeful of tying down Gabriel Barbosa for another two years on top of his current deal, extending his deal to expire in 2021 instead of 2019.
New boss Antonio Conte took charge of his first training session on Wednesday, despite the club missing some players who are resting after being on EURO 2016 duty, and is keen on getting the majority of his business completed before their preseason trip to the USA.
Chelsea have already signed Belgium striker Michy Batshuayi for £33million, but it appears the Italian manager wants more firepower to rival Diego Costa.
Barbosa has scored twice in just four national team appearances, including netting in the 7-1 rout over Haiti at the Copa America.