Set photos from the upcoming Batgirl movie seem to indicate that there is a Robin active in that version of Gotham City. It's still uncertain as to how Batman will even be handled in that movie - Commissioner Gordon will be played by The Snyderverse's J.K. Simmons, but Ben Affleck will have already retired from Playing the Dark Knight following the events of The Flash - a film which will also bring back Michael Keaton's Bruce Wayne, setting him up to appear in Batgirl. Maybe all of this will make more sense once we se a trailer, but it seems a bit crazy right now.
Batman met with The Avengers a few times way back in my Brave and the Bold: The Lost Issues! days - First when he tried to rewrite the team's roster, then when he was a part of the the original line-up and finally with Cap's Kooky Quartet...