The combative midfielder's present deal expires next summer and there have been murmurings he would leave before the start of the new season.
Everton manager David Moyes has been keen to discuss the situation with Gravesen and finally held talks with the Denmark international in the United States.
But while Gravesen informed Moyes he is prepared to stay on Merseyside, he wants to see a marked improvement in the team before he inks a fresh deal.
"I have spoken to Thomas about the situation and he told me that he wants to see Everton are making progress," Moyes told the club's official website.
"But he also said that if we do that he would have no hesitation in staying."
Gravesen's team-mate Gary Naysmith appears to be more inclined to remain with The Toffees, irrespective of the club's plans to strengthen the squad.
Naysmith is also entering the final year of his contract and he is looking to open discussions regarding an extension.
The Scottish left-sider said: "I have less than a year on my contract to run and hopefully in the next two or three weeks my agent will be going to meet the board and try and get a new contract.
"There's been a lot of speculation in the papers about other teams but I haven't heard anything about being swapped or being sold to any other team.
"I have a lot to prove this season because I have to get a new contract and the best way to do that is to be in the team."
