Fernando Alonso was previously allowed to compete in the 2017 Indianapolis 500 while at McLaren, and even after leaving F1, the Spaniard made two further entries with the team.
In 2018, after being named as the official test and reserve driver for McLaren, Norris contested the Rolex 24 at Daytona with Brown's United Autosports team.
Since his 2019 promotion into a full-time race seat, however, the Briton has remained focused on F1 alone.
Asked if his new "multi-year agreement" related exclusively to F1 or extended to the Indy 500 and Le Mans, Norris grinned and said: “Good question. I’m not going to say too much.
"It’s definitely something I would still love to do at some point in my career. It’s not going to be this year or next year.
"This year, with Formula 1 with Covid and everything, you’re just not really capable of doing that and next year with the new rules, you want to be spending the time with the team and focusing on one thing.
“If you’re not, then you’re not doing a good enough job and you’re not doing everything you can to maximise your one objective."
Although the chances of tackling a non-F1 race are a few years away, Norris stated both he and Brown have the same targets of extending their partnership into other categories.
“There are definitely things I would want to do and Zak has the same opinion and objectives," added Norris. "We would love to do more races together like we did in Daytona with the 24 Hours.
“I would love to do more endurance 24-hour races or 12-hour races, whatever it is. I had a lot of fun doing those kinds of things so it is definitely something that is in the back of my mind in the future.”
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