You know what would be cool?
The Flash having a friendship with Superman that is almost solely today's Flash hanging out with the Superman from like 500 years from now in a continuity where Superman doesn't age or barely ages (Maybe some gray around the temples).
Think about it. All of Superman's friends from this era are dead or gone by then, but there he is having barely aged. The Flash of today periodically winds up in that future for various reasons. Every time he does, he looks up Superman and they hang out (Fortunately he'll wind up going back to the future always like a month or a year or a decade after the last visit, so it'll be chronological for both characters) and eventually become best buds in 2531 AD or whenever kind of by default because he is literally the last hero standing who is contemporary to Superman's first century of life. Superman stays vague on what happened to the others, just that they are either "dead or... gone" to avoid accidentally changing history (and leave open storyline possibilities- you know, so and so may not be dead, but rather hanging out on the other side of the galaxy or whatever. It can be left ambiguous.). Still, he's delighted to see The Flash whenever he shows up because The Flash is now his oldest friend! The whole dynamic is totally different.
Then, The Flash has to return to 2021 AD where he and Superman aren't that close, and Superman is sort of one tier above, and they just do the Justice League and whatever else together, mostly just co-workers as superheroes. And, The Flash, being the Flash, constantly slips up with in-jokes and stuff that he shares with future Superman and then goes "Aw, crap. Spoilers. Sorry. Remember this, you'll find it hilarious in about 500 years or so when we're best friends..".
The claim that they are best friends in the future can be something that good naturedly annoys Clark, but he can't actually refute it because he hasn't lived those years yet.
It seems like it could work with either Barry Allen (That's honestly who I'm picturing it with) or Wally West, but probably less so with Jay Garrick- Jay Garrick seems more like he'd be Superman's very serious friend from the Justice Society in an alternate continuity's 1940s or 1950s. Allen or West I picture getting a laugh out of Clark and loosening him up in a future that has by implication been kind of hard for him, sort of standing sentry for earth for all that time while everyone around him has died or moved on to something else (Sure, that time has it's own newer heroes and he works with them, but they're all a few centuries younger than he is and all either think of him as an idol who they grew up wanting to be, someone out of touch with contemporary (future) society, and/or as a father/grandfather/pioneer figure rather than as straight-forward equal and friend.). Flash keeps him optimistic and hopeful through humor and fun, a contemporary still alive thanks to time travel, and any sense that he's not an equal is something Clark has long since changed his mind about- in a way he's the only one who just sees Superman as Superman and who Superman can trade late 20th and early 21st century references with (Either to the era's popular culture or to the superhero heroes, villains, and adventures of that era).
I don't know if it'd work right *now* in the comics because of what they are doing with Clark seemingly aging a lot faster than this sort of recurring story point would age him, with Jon hanging around and whatever (Maybe Jon could be dead- he's half-human, he could age faster. Maybe that's why Superman is down 500 years from now. Jon died a century beforehand and that was his last connection to his early days as a hero, what would have been Clark's human lifespan. Sure, he's got descendants, but it's not the same.). However, the next time they reboot continuity, in an out of continuity series of comics, or in some alternate media like television or movies, it could be a thing.
I guess the other way to go that would work with current continuity a little better would be that The Flash from the main DC universe is best buds with the Superman from, I don't know, let's say Earth-52, and basically the same dynamic plays out as above where the main universe Superman and he don't have the same relationship as Flash has with this other Superman, but Flash occasionally gets them confused and then has to go "Sorry, inside joke with the other Superman" when teaming up with the main DC universe Superman.