Batman has been referenced in three of the universes
Earth-1 (main universe)
Bruce Wayne exists in this universe, is will known, and lives in Gotham. There is no confirmation of him engaging in vigilante activities.
Additionally, the future newspaper shown in the Flash mentioned a "Wayne Tech" company, which may possibly be associated with Bruce Wayne.
Oliver Queen: Photos can be doctored. They could have put Bruce Wayne's head on that body. Has Bruce Wayne left Gotham to hang out in Star City recently? No.
Arrow season 6 episode 2, "Tribute"
Earth-2 (Flash alternate universe)
A character by the name of "Bruce" was found on Earth-2's Barry Allen's phone alongside "Clark" and "Diana". Nothing else was said about him. (Note that Earth-2 Barry doesn't have the speedforce.)
Earth-38 (Supergirl universe)
An unnamed vigilante matching Batman's description was said to have once worked with Superman.
My cousin worked with a vigilante once, tons of gadgets, lots of demons. Vigilantes are nuts!.
Supergirl season 2