Depp, 49, has apparently bought the bisexual beauty a horse so they can go riding together on the set of his new movie "The Lone Ranger", reported Sun online.
"Amber's been a regular visitor to the set. Johnny's even bought her a horse so they can go riding whenever he gets a break," a source said.
The rumours spread when the two filmed "The Rum Diary" in Puerto Rico.
Heard, 26, who is openly bisexual and had been in a relationship with Tasya van Ree until last year, is said to have even danced for him seductively on the set of the 2009 film.