In the Terror storyline of Legends of the Dark Knight, he is in a "relationship" with a female mannequin wearing lingerie and a cowl like Batman's.
Bruce's childhood friend-turned-villain Thomas Elliot, who became Hush.In Shortpacked! it's mentioned "Even straight dudes dig Batman.In Death of the Family, Batman has an internal monologue about how the Joker's eyes never change, never revealing anything about his thoughts or emotions.He outright says Batman has a cute little ass in Arkham video games. Several canon Batman comics have the Joker professing a sexual fascination with Batman, to nightmarish extents. The pages are lined with homoerotic content and an unnatural level of sexual tension.
In the macabre Arkham Asylum: A Serious House on Serious Earth, which is a canon Batman comic, Joker slaps Batman on the butt and continually hits on him throughout the comic. Harley and Ivy both think that the Joker only has eyes for Batman (and Harley is in love with the Joker) and the Joker has often professed affection towards Batman (complete with moments of desperate attention-needing).He's been romantically involved with multiple women, but also attracts a lot of Foe Romance Subtext with his male enemies.