The 13-time grand prix winner said he has seen “a different Lewis in this challenge” and believes the Mercedes driver is “so motivated” by the title fight.
Red Bull leads the constructors’ championship with 286 points ahead of this weekend’s race at Silverstone, leading Mercedes by 44 points, while Verstappen’s 182 points in the drivers’ standings puts him 32 clear of Hamilton.
Coulthard, who is also the W Series Advisory Board Chairman, also said he believes Verstappen and his team have “extended” Hamilton’s F1 career.
“What I’ve observed, if you look at the last couple of races which he’s lost, he’s been much calmer and much more magnanimous in defeat than we’ve seen,” Coulthard said ahead of this weekend’s British Grand Prix.
“Sometimes when he was beaten by Nico when they were teammates, he would be a bit chippy, he would be a bit dummy in the dust and a bit spoiled child, multi-millionaire racing driver.
“I’ve not seen that. What I’ve seen is an elegance, I’ve seen a confidence, I’ve seen a different Lewis in this challenge.
“So therefore, I think that he is truly relishing the challenge. Of course he wants a title, who doesn’t? Of course he wants to win, who doesn’t? But he’s done so much winning.
“I think that the fact that he’s signed up for another two years, he is just so motivated by the challenge.
“I think that Max Verstappen and Red Bull this year have extended his career, I truly believe that, because if he doesn’t win it this year, I don’t think he’ll go off into the winter crying his heart out.
“I think he’ll go ‘you know what, as a team we didn’t deliver, how can we do better next year?’
“I just think he’s such a pure-bred racer – the way he’s been brought up by his family, total focus, total dedication.
“I think he loves it, I truly do, and I think the fact that he’s not winning right now is extending his career.”
Coulthard also said it feels “feels a little bit like Mercedes have to deliver this weekend, it feels like Lewis has to deliver this weekend” after Verstappen won the past three races – one in France and the double-header in Austria.
“I’m really curious, because Toto obviously has a belief in the performance of the car,” he said.
“The team have done so much analysis and understanding of where their cars are performing well relative to the competition that they will have run the sims and will know reasonably accurately, taking out weather and various other factors and variables, where they can expect to perform.
“So it feels a little bit like Mercedes have to deliver this weekend, it feels like Lewis has to deliver this weekend, because Max, he’s not scared of anyone, he is a natural-born winner, he is on the crest of a wave right now, and he’s not going to be fazed by lots of Union Jacks.
“But I do think that Lewis is someone who does respond in a kind of Nigel Mansell way to home support and truly believes it can lift him.
“I think it’s a really important weekend for the championship.”
The third round of the 2021 W Series season at Silverstone is live on Channel 4 this Saturday from 12:05 GMT.