Verstappen, who strolled to the title last year by winning 19 of 22 races, has opened the door to his rivals as he has now gone five races without victory.
Norris' win at the Dutch Grand Prix last weekend, his second this season, put the Briton to within 70 points of the Red Bull triple world champion with nine races remaining, but he is ignoring the hype going into this weekend's Italian Grand Prix.
"For me, a win is incredible and is generally, I think, one of the best feelings you can have, because this is what I dreamed of and this is all I wanted to achieve," Norris told reporters.
"I really do not think of the championship at all, for me, it's just one race at a time.
"I couldn't care less about next weekend or the weekend after or whatever. It's just what can I do today? What can I do tomorrow? What can I do Saturday and Sunday? That's just how I live."
Norris, who also won in Miami in May, is determined to keep up the pressure on Verstappen at Monza, but believes the constructor's championship, where McLaren are 30 points off Red Bull, may be the most likely chance of success.
"We know we've been performing well the whole year since Miami. We've been doing a very good job, and this was a weekend where everything just went perfectly," Norris said.
"Of course our goal is to catch them in both, and especially from a constructor side, it looks a lot more doable than the driver's side.
"But I'm doing my best, the team's doing their best, and that's all we can hope for."
