The governing body has controversially updated its International Sporting Code ahead of the 2023 season to require drivers to seek FIA permission to make personal statements.
A new clause says a breach of the rules now extends to "the general making and display of political, religious and personal statements or comments notably in violation of the general principle of neutrality promoted by the FIA under its statutes".
This has generated significant backlash, with Red Bull team principal Christian Horner, saying that suppressing drivers' voices and opinions risk turning the grid into a "load of robots".
Likewise, McLaren driver Norris says the FIA is treating drivers like they are in school instead of trusting them to be "grown-up" but reckons significant criticism of the amendment may force a U-turn.
Speaking at the launch of the MCL60 on Monday, the Briton said: "There are things that you want to do, that you're going to want to say, which maybe they don't allow it.
"I feel like there's been quite a bit of pressure and enough said to maybe make a little bit of a U-turn.
"But F1 have made things clear with what they think is acceptable and what we should be able to do as drivers, and that's what I stand by.
"We should be able to say what we want and what we believe in. We're not in a school.
"We shouldn't have to ask about everything and say, 'Can we do this? Can we do that?'
"I think we're grown up enough to try to make smart decisions. Maybe sometimes people make silly decisions, but that happens in life."
Norris underlined that clamping down on driver statements would undermine the platform F1 drivers have to make a positive impact.
He continued: "We're doing it because we have a lot of fans, millions of fans, millions of viewers who we want to influence and guide and use to help or to help them personally.
"We should be able to say and do what we want. That's what defines people, is what creates us, what makes us human.
"We're only trying to help people in the world and give advice and so on, and there's no reason why we shouldn't be allowed to do that."
Last week, FIA president Mohammed Ben Sulayem sent a letter to F1 team bosses informing them that he has stepped back from the day-to-day management of F1.
However, this move after the first 14 months of his term, was pre-planned and not a direct response to the fractious relationship with F1 – which deteriorated further in recent months after Ben Sulayem said the value of the series was overinflated, called out the grid for its opposition to an 11th team and the resurfacing of misogynistic comments.
