The young Toffees forward enjoyed a breakout season in 2021-23, making 40 appearances, scoring four goals and contributing three assists.
He has become a key part of Frank Lampard's side at Goodison Park and appears to be attracting interest from Stamford Bridge.
The Times have claimed that Everton deem Gordon not for sale, although a bid of around £50 million for the player may change that stance.
The Toffees are interested in Chelsea striker Armando Broja and Thomas Tuchel's side could use that as a bargaining tool in a potential pursuit for Gordon.
There has also been interest in the forward from Tottenham Hotspur and Newcastle United as well.
Spurs wanted Gordon to be part of a joint deal to bring the young Englishman and Richarlison to north London. Everton rejected this proposal with just the latter joining in a £52.2 million.
Everton are looking to tie Gordon down to a longer deal and continue his development on Merseyside.
He has three years left on his current deal with Lampard's side.
Having lost Richarlison, Everton will be hugely reluctant to allow another forward to depart this summer.
Chelsea's interest stems from a lack of attacking options up front for Tuchel's side following the departures of Romelu Lukaku and Timo Werner.
The Belgian striker has rejoined Inter Milan on a season-long loan whilst the German has returned to RB Leipzig in an £18 million deal.
Kai Havertz has been used in a false 9 role but other than Broja, there is no recognized striker at the Bridge.
Chelsea star Havertz on taking up the striker's role this season
Havertz has been playing as Chelsea's main striker throughout the pre-season and in the side's opening two games of the Premier League season.
He has spoken about taking up the position having previously been an attacking midfielder, telling the club's official website:
"I've played a lot of positions over the years but last season I played a lot as a number nine and it's a position I like very much. If I look back four years ago, I could never have imagined playing there. But the last two years I've played it more often and I like the position."
"It's still good that I am flexible and can play different positions up front, but I am there to score goals and make assists and as a number nine you have to do that and link up with players."
Although Havertz wasn't on the scoresheet against Tottenham on August 14 in a pulsating 2-2 draw, his performance was one that impressed.
But Tuchel's side seem keen on bringing Gordon to Stamford Bridge to add more goals from attack.