The sport's governing body, the FIA, has delayed its findings into whether any of the grid's teams have broken the financial rules until Monday.
It was anticipated that the FIA would reveal on Wednesday those that failed to comply with last season's £114million cap.
Red Bull faced unproven allegations at the Singapore Grand Prix that they exceeded F1's costs cap last term - bringing into the spotlight the legitimacy of Max Verstappen's already controversial championship win over Lewis Hamilton.
With the decision now being delayed, Norris believes it will hang over the Japanese Grand Prix and, when asked if punishment needed to be swift and severe, the McLaren driver replied: "Of course.
"I think because teams are trying to find and trying to be as efficient as possible with every single thing that they do, any small amount over is, in any part of Formula One, maybe not huge, but depending on what it is, a big advantage.
"Considering there has been such a rule implied to make things a bit more comparable for everyone, give all teams a better chance at fighting, then it's just an unfair stat within the performance of Formula One. There should be a pretty hefty penalty for whoever does cross the line."
Speaking on Thursday, Hamilton said it was "imperative" that the correct punishment was doled out while the likes of Charles Leclerc and George Russell echoed those comments.
Norris feels the budget cap has suited McLaren over others in the pit lane who were spending more and that any team who had breached the limit could take years to catch on track.
"I think for us, it suited us very well in where we were performing and the budget that we had relative to the bigger teams," added the Brit.
"For those bigger teams it has been about choosing where to place that budget into, whether it's more performance or aero or design - all of this stuff.
"It's not just literally a direct performance impact on track, there can be many other ways around it, which you can benefit and the thing is, it doesn't just benefit you for one single season.
"If it helps you in that first one to get going, it's always kind of a knock-on effect the next few years.
"If it is true, I don't know what is true or not, but if it is true then yes, it can help someone focus more last year if that's the case, and then put more people, more development on this year.
"If you can start a new era off on the right foot, then it's only going to keep you ahead for a good amount of years until kind of everyone starts to catch up, especially so early on, then it would make a big difference."