Blues head coach Thomas Tuchel is keen to strengthen his squad with three signings before the transfer window closes next month.
Gordon, who has also attracted interest from Tottenham and Newcastle, is viewed as an exciting prospect for the future who could fill one of the three attacking positions in Tuchel's preferred 3-4-2-1 system.
Everton are continuing to insist the 21-year-old is not available for transfer and Chelsea will now consider whether to make an improved offer.
Gordon started both of Everton's Premier League matches this season, including an opening weekend defeat to Chelsea at Goodison Park. The England Under-21 international made his top flight debut in January 2020 and has so far made 51 Premier League appearances for Everton, scoring four goals.
It remains to be seen whether Chelsea consider making a player-plus-cash proposal in any improved offer.
Everton have shown some interest in signing Chelsea striker Armando Broja, who spent last season on loan at Southampton while manager Frank Lampard is also an admirer of Blues midfielder Billy Gilmour.
Chelsea are continuing talks over moves for Barcelona forward Pierre-Emerick Aubameyang and Leicester City defender Wesley Fofana. The club will also move for Frenkie De Jong if the Dutchman decides to leave Barcelona.