[SPOILERS for those not caught up on Supergirl ahead.]
Early in season 2, Supergirl surprised fans everywhere when it introduced their its own live-action version of Superman/Clark Kent (Tyler Hoechlin), something that seemed unlikely to happen in The CW's DCTV universe anytime soon. Luckily, the show's risk paid off well, with both fans and critics instantly taking a liking to Hoechlin's Man of Steel, leaving open the possibility of him returning down the line. The show's creative team even went on to confirm the existence of Gotham City (and therefore, possibly Bruce Wayne/Batman) on Kara's Earth, leading many fans to speculate about whether or not we will ever see a DCTV version of The Dark Knight.
Since then unfortunately, no other comments or references connecting back to Batman have been made in any of The CW's DC TV shows. That is, until the episode 'The Darkest Place', when Supergirl not only confirmed the existence of Gotham City, but also seemingly, directly confirmed the existence of Batman in the DCTV universe (or Arrow-multiverse, if you prefer).
The reference came near the beginning of the episode, while Kara was discussing her distaste for vigilantes with Alex (Chyler Leigh), Jimmy (Mehcad Brooks), and Winn (Jeremy Jordan), as the latter two continued to be secretive about their crime-fighting adventures as Guardian in National City. To prove her point, Kara referenced a particular moment in her cousin's superhero career, when he once worked with a vigilante:
“My cousin worked with a vigilante once. Tons of gadgets. Lots of demons. Vigilantes are nuts.”
It's possible that Kara could be referencing another vigilante from the DC comics, though this description certainly seems fitting of Bruce Wayne more than anyone else. Following the Gotham City reference earlier this season, this opens up even more exciting possibilities for an introduction of Batman sometime down the line. DC Films and Warner Bros. in general also seem more open now to the idea of letting their larger superheroes play around in the sandbox that has become their TV universe.
What this doesn't confirm is whether or not there's a Batman on Oliver Queen (Stephen Amell) and Barry Allen's (Grant Gustin) Earth, since Kara exists in a separate universe from her fellow DC heroes and vigilantes. It's entirely possible, in fact, that there's only a Batman on her Earth, meaning we could only see Kara interacting with The Dark Knight onscreen, assuming she ever even gets to.
On the other hand, these Batman references might not be the type that the DCTV creative team plans on paying off sometime down the line - and thus, fall more in line with the Green Lantern and Aquaman references that have been featured in both The Flash and Arrow before. While those are fun, they haven't resulted in those DC characters being introduced in the Arrowverse (and probably won't, what with the DC Extended Universe up and running now). So basically, when it comes to Batman appearing in Supergirl, best to not hold your breath, just in case.
Supergirl continues Monday November 28 with ‘Medusa’ at 8pm on The CW.