The 31-year-old is in the last year of his current deal at Goodison, but is eyeing up an extension.
The versatile Ivorian attacker has had his injury problems since moving to Merseyside from Wigan and has something of a forgotten man for an extended period, but looks back to his best this season after regaining fitness.
Kone has popped up with some important goals for Roberto Martinez’s men this term and has started five of the club’s first eight Premier League clashes in 2015-16.
The African attacker has stated that he is working hard to convince the club to extend his terms past this summer and is not keen to leave the Toffees at the end of the current campaign.
“I don’t feel any pressure that it’s the final year of my contract. If anything it’s a positive pressure that I see as adding to the challenge,” he told The Liverpool Echo.
“When you sign for a club and almost immediately you’re out for nearly two years the fans think ‘is he not fit? Has something else happened?’
“This is the first season that I’m 100% cured fitness-wise and back to my best. It’s all about making up for that last time whether I stay or whether I go.
“I’d love to stay and extend my contract. Even if I do go I’d like to leave a lasting impression that here’s a guy that when he was fit, he presented a good image of himself at Everton.”
Kone has worked well in tandem with Romelu Lukaku this season, with the pair linking up well and causing opposition defences considerable problems.
The veteran has been used in a number of positions but has started the last couple of Premier League games from the Everton bench.