Button has previously made no secret of the fact he wants to venture into sportscars at some stage of his career, with Le Mans high on his agenda.
Given McLaren have a GT programme up and running after the recent release of their top-of-the-range sportscar, the MP4-12C, running at Le Mans could be an option in the future.
The only problem for Button is being given permission to compete in such an event while he remains contracted to McLaren in Formula One.
"I'd love to do that (Le Mans) next year. I'd like to give it a go," Button said.
"I'm not sure I'd be allowed, but I'd love to. I tried to do a 24-hour race back in 1999, but the car broke down before I got in it!
"But yeah, I'd love to give Le Mans a go, although I don't know how happy they (McLaren) would be about that.
"And I've heard when you are in a GT car it's probably the worst experience at Le Mans because you are just looking in your mirrors the whole time with these things (the prototypes) coming past you like you are stood still.
"I don't know, but it's something I'm very interested in."
Whitmarsh, however, has made it clear Button will have to put such plans on hold.
Asked whether he would allow Button to take part in Le Mans next year, Whitmarsh said: "Next year? Probably not. He hasn't asked me.
"We don't plan to have a McLaren next year, but we'll go back there one day because we have happy memories of being in Le Mans, and one day in the future he (Button) might be doing that.
"But over the next few years I think he'll be concentrating on F1 and not Le Mans, but I'm sure we can accommodate it in the future."
Robert Kubica's accident in a rallying event before the start of the season highlights for Whitmarsh the dangers of a driver being allowed to take part in another form of motorsport when they are contracted to a team.
Whitmarsh added: "I do consider things like that and I normally remind them of such incidents.
"It's the nature of drivers that, in a very healthy way, they feel invincible, that they will never hurt themselves, and they have a passion to go and do these sorts of things.
"It happens reasonably frequently that drivers come along and ask to do something.
"But there is already risk in Formula One, so we are not usually wildly enthusiastic about introducing others."