Chelsea are determined to land the player, having already had £20m and £26m offers rebuffed by the Merseyside club.
And Sky Sports now claim an improved offer has been waved away by bosses at Goodison Park.
Stones, 21, boasts four England caps and has started both of Everton's two Premier League games this season.
Chelsea stepped up their pursuit of the youngster after losing 3-0 to Manchester City on Sunday in a game that saw veteran captain John Terry subbed at half time.
A source close to the player said: "JT likes a fight and loves playing for Chelsea. Don't expect him to walk away from this one.
GETTY
Everton value defender John Stones at around £40million
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"He'll continue to play his heart out and is convinced he can prove his value to the manager. Nobody will work harder than him in training this week.
"JT knows other clubs would line up to take him but he is dedicated to Chelsea and can't see himself wearing any other shirt. He expects to be in the team to play West Brom and is confident Chelsea will win the title this season with him playing a key role as captain."