The Woking team are fifth overall but only 11 points behind Aston Martin after taking a 47-point haul from Qatar's sprint weekend.
To overhaul Ferrari, the only team to have beaten dominant champions Red Bull so far this season, McLaren would need to score 80 points more than the Italians over the final five rounds.
"It’s doable. I think so. A couple of races ago it was 70-something (points) to Aston," Norris told reporters after finishing third in Sunday's Qatar Grand Prix with Australian rookie Piastri second.
"I think there's going to be some races that are going to be more difficult than here and probably will suit other teams a bit better," said Piastri.
"I think 79 points in what, five races and a couple of Sprints? It's a big challenge but yeah, we'll give it a go. I think obviously the gap to Aston now is very achievable. Hopefully we can do it."
Red Bull's race winner Max Verstappen, who clinched his third title at the Lusail circuit in Saturday's sprint won by Piastri, said former champions McLaren were now looking like the most consistent rivals.
"I do think they have the best driver lineup out of all of them," he added. "They are operating really well."
Piastri's sprint win was something of a breakthrough for the 22-year-old who has been getting better and better as the season goes by.
It was his first win, albeit not a full grand prix, and followed his first fastest lap in Italy and first podium in Japan two weeks ago.
The Melbourne driver, guided by former Red Bull racer Mark Webber, is coming through strongly as a youngster capable of taking the fight to Verstappen, who is still only 26.
Piastri might have raced for Renault-owned Alpine, had he not very publicly decided otherwise last year, and he will not be regretting that decision with McLaren now a hefty 129 points clear of them.
The team that started the season with five complete no-scores in the opening eight races have come alive since Austria in July, where they introduced significant upgrades to their car.
No team, not even dominant Red Bull with their 15 wins in 17 races, have scored as many points in the last three rounds and McLaren are also the only team to have been on the podium in the last three.
That did not stop Norris from beating himself up about his performances.
"I should have fought for two pole positions this weekend. And potentially two victories," said Norris, who has yet to take a first win.