EXCLUSIVE

Johnny Depp could have saved the paint — and the blood — when he allegedly wrote the words “Billy Bob” with what was left of his finger on a mirror … because Mr. Thornton says he never hooked up with Amber Heard.

TMZ broke the story … Amber and her people claim in March, 2015 Johnny accidentally cut his fingertip off in the middle of an argument with Amber … and did his artful handiwork on the mirror after accusing her of cheating with Billy Bob when they co-starred in the flick, “London Fields,” which was set to be released in 2015.

Billy Bob tells us the cheating allegation is ridiculous … “completely false.”

Thornton says he never socialized with Amber off set, other than a cast dinner. Even on the set, he says he didn’t hang with her … he was mostly around members of the crew. He adds he didn’t even have a friendship with the actress … it was all professional, on the set, and that was it.

As for Johnny, even though they were on the same set, Billy Bob says they had no interaction … did not even speak.

Billy Bob’s been married to wife Connie Angland since 2014. He says he’s talked to Connie about the story and assured her it was Johnny’s fantasy.