Already the captain of England's ODI and T20 sides, Brook would be an obvious choice to lead the Test team, too, given he has been the vice-captain to Stokes since September.
England's next Test series is against Pakistan later this summer. They host the Ashes in 2027.
Stokes retired from international cricket on Monday after the Test series loss to New Zealand, in which Brook was also involved.
The following day and speaking in his role as T20 captain ahead of the five-match series against India starting on Wednesday, Brook said it would be "great" to be Test captain.
"It'd be a privilege to do it, to captain England in the highest format of our game. t's the pinnacle.
"Playing Test cricket is the greatest thing that I've ever done in my life and it's a dream, something that I've always wanted to do since I could speak.
"It's not up to me, that decision. But if I got offered it, then I'd be happy to take it."
Brook accepted it would be a "tough job" to juggle being captain in three formats but said it would be manageable because he has "committed completely to England cricket".
"I've said that I don't want to play any franchise cricket, bar The Hundred, and everything that I want to do is to play cricket for England. And whatever I do on and off the field is to try and perform as well as I possibly can for England.
"Hence the reason I don't play in the IPL and PSL and all the other franchise competitions."
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