The movie, an adaptation of the late Thompson's novel, stars Depp as the boozing Paul Kemp, a writer who works as a journalist in Puerto Rico.
Speaking with Radio Times, Depp said that he found the movie's manuscript in Thompson's basement.
"I remember when I unearthed the manuscript out of a box in his basement. He didn't know where it was and I pulled it out and he was like, 'Jesus Christ'," Depp said.
He added: "We read it together at three in the morning and it was, 'We should do this together', but Hunter didn't exactly fill his end of the deal. I guess he had other things to do."
Depp praised director Bruce Robinson as a "genius" for his work helming the picture.
The Pirates of the Caribbean: On Stranger Tides star also revealed that the movie's cast and crew participated in a daily ritual on the set to honour Thompson's memory.
Depp explained: "Each day on set we had this ritual [to honour Thompson's memory]. Hunter would have his own chair right next to Bruce's and he had a script with his name on it, the whole thing, and we would put a pack of Dunhills on a little table by the side, a bottle of Chivas, his favourite drink, and a tumbler full of ice.
"And each morning we'd dunk our fingers in the tumbler and dab a little Chivas on our necks, like a sort of Hunter cologne, take a sip and then off to work we went. We did that every morning, so Hunter was always there. It was good."
The Rum Diaries will be released by FilmDistrict on October 28.