With two race wins from two before the summer break, Lewis Hamilton and Jenson Button have closed the gap to championship leader Sebastian Vettel. Hamilton, in third place, is now 88 points behind the German with Button a further 12 back.
Whitmarsh admits that with only eight races left in the season it will take a monumental effort to catch Vettel but his team certainly haven't given up.
"I think it's though, but it's possible," Whitmarsh told Autosport.
"And winning races is tough, but I think all you can do is try to win every race, and you can't do more than that. It is difficult to pull back the gap, but it's possible.
"Of course people focus on the championship, but I think if you are not leading the championship then you can't worry too much about it.
"I'd like a one-two and it'd be handy if a Red Bull was not the third car on the podium if we get a one-two, and so we'd welcome Ferrari on the podium as well. We never give up. We've never been a quitter and this time isn't one.
"Whatever happens we are going to try to win every race and that's the great challenge."
Hamilton has come under heavy criticism for his behaviour on and off the track this season, with his aggressive driving style and at times controversial comments leading to the 26-year-old to take strain under a constant barrage of negativity.
However, Whitmarsh reveals that Hamilton is back on the right track, with his mind firmly focused on winning his second World title.
"I think the positive one is that his head's now in a great place, because he suddenly believes he can win, and there's lap time in that," explained the Briton.
"I think Lewis desperately wants to win. I think he's tough on himself. He still takes too much to heart what's said in the media.
"He has a little bit too much sensitivity about that. But he will learn. At the same time he is a young man, he gets a lot more attention, a lot more pressure, a lot more expectation.
"We are trying to help, we are trying to work with the media. I'm probably a bit more cynical than Lewis, but the same time I probably don't get quite the battering that Lewis usually gets," he added.