Both Button and Hill have voiced the opinion that Hamilton could lose interest if he finds himself with an uncompetitive Mercedes for a second consecutive season.
Hamilton’s contract runs out this year and, inevitably, speculation about his long-term plans has already begun, even before the first race of the season.
When asked about the comments made by Button and Hill, Hamilton said: “Ultimately people creating rumours without facts, it's never helpful.
“And you would have thought that they were both know by now. I've been with Mercedes since I was 13. Having [had] a difficult year [like] we had last year, I'm still here. Whether or not we have a difficult year this year, I'll still be here.
“I'm a fighter, and we fight as a team. I love the challenge of finding solutions and I still believe that I'm able to put the car in places that perhaps others are not able to. And I love that challenge.
“Of course I wish to be starting the season with a great car. But it's the journey I think that really counts. So there is no hold up with our contract.”
Hamilton insisted that he has plenty of time to agree a new deal.
“I've always been very, very relaxed. I don't feel like I have to get it done right this second. I'm in a very fortunate position. It'll get it done when we are ready.
"I have a great relationship with Toto [Wolff] and with Mercedes. And we fully support each other. And I'm really excited to build the future together.
“I'm really proud of the work that we're doing on and off the track, and the potential new things that we can do moving forward. So we'll get there - unless something happens with our relationship and me and Toto get into the ring! Otherwise, we're good."
Hamilton was also asked if he thought Red Bull rival Max Verstappen might lose motivation having already won two world championships.
"I can't really comment on Max,” he said when reminded that in 2015 team-mate Nico Rosberg seized the initiative at Mercedes after Hamilton had clinched the title.
“I think it's always possible as an athlete, of course, once you've had it sealed, the pressure does switch, and it's different. You can sleep more, you don't have to push in training as much, because you have different deliverables.
“But I think he will be very, very confident. I think they developed an amazing car last year, they blew away all the records on pretty much everything.
“And I don't even think they were pushing at the end and still were way ahead. I don't see that changing necessarily start of the season, so they can be quite relaxed hopefully.
“But hopefully, those that are just behind will continue to apply pressure, I think Ferrari have got a decent package, Aston have got a decent package and we are hunting too."
Pushed on Verstappen, he added: "I don’t think he’ll slip up. He's a world champion. And so I wouldn't question his determination or his focus. I think he will be just as focussed as ever, and it's our job to catch up.”