Madonna goes way back with Billy Idol ... she says she's known him since he was swiping her taxi cabs.

The pop star recently told Bilt she and Billy both lived at The Music Building during her early days in New York ... and they had anything but a meet-cute.

Taxis are a hot commodity in the Big Apple, and Madonna recalled a time she had just flagged one down ... when Billy himself swooped in.

Madge recalled ... "He steps in front of me with his punk, like, regalia ... He didn't even say 'sorry' or 'thanks' or nothing." She said the interaction left her wanting payback.

Madonna told the funny story as part of her collab with Bilt -- a loyalty and rewards program -- and she's helping pay next month's rent for musicians currently leasing space at The Music Building.

She also worked with the fintech company on an exclusive "Confessions II" vinyl, available exclusively for Bilt Members.

No word if Billy's getting a copy ... but something tells us Madonna's not going out of her way for him.