There was one hinted at in the DC Animated Universe cartoons, which was confirmed in associated comics.
A 1999 episode of Batman Beyond1 implies a past relationship between the two:
Barbara: Dick finally got fed up living in Batman's shadow. He decided to leave. He was hurt when I chose to stay behind with Bruce.
Terry: As his partner. (Gordon's grin widens.) His girlfriend? (Gordon's grin widens even further, leaving McGinnis shocked.) Whoa!
Batman Beyond Season 1 Episode 12: "A Touch of Curaré"
This relationship is also hinted at in other DCAU properties, most notably in the TV movie Mystery of the Batwoman2:
Barbara: I just saw the news and was wondering if you got a new partner. Someone a little older?
Bruce: I have no idea who she is.
Barbara: Because if you had, I'd be really upset...because I thought you and I were, you know...
Batman: Mystery of the Batwoman (2003)
The comic series Batman Beyond 2.0, released in 2014, explored the relationship in a little more detail, revealing that Bruce and Barbara had an intimate relationship while Dick was in Blüdhaven, and that Barbara got pregnant:
Barbara: I'm seven weeks pregnant.
Bruce: [beat] I'll have to congratulate Dick.
Barbara: He's only been back for three.
To the best of my knowledge, there has never been a romantic relationship between Bruce and Barbara in any of the mainstream comic continuities.
1 For the uninitiated, Batman Beyond was part of the DC Animated Universe, a shared continuity of cartoon shows and TV movies that aired between 1992 and 2006. Batman Beyond was a sequel to the wildly popular Batman: The Animated Series (and its direct sequel, The New Batman Adventures), and is set about 40ish years later
2 Thanks to starpilotsix for reminding me of this in comments
