The actress is reportedly looking to 'settle down and have a family' with the tech entrepreneur.
"Amber and Elon are both very serious about each other. She would love to get married," Amber's father David said to Grazia magazine.
David added that one of the things the couple wants to do is settle down and have a family, saying that they are making plans for that.
The 30-year-old Hollywood star was first linked to Musk last summer after the two of them were spotted together on several occasions in London and Miami.
Amber is said to be 'smitten' by the 45-year-old Tesla boss, following her divorce with Depp.
According to a source that spoke to the Mirror in January, The Rum Diary actress is "over the moon about her divorce but is even happier about being able to go public with Musk."
The couple is said to be already planning a lot of adventures together, and Amber is relieved she can move on with her life.
Musk whose fortune is said to be an eye-watering $9.2 billion is said to have become 'infatuated' with the actress after the two appeared in the 2013 film, Machete Kills.
Despite the pair not getting a chance to meet on set, the billionaire is said to have emailed director Robert Rodriguez repeatedly in the hope a meeting between him and Amber, despite the fact that she was still with Depp.
Amber filed for divorce from Depp in May. Depp was accused of abusing the actress, but he denied the allegations.
Their separation was finalized in January after Depp agreed to give over $9.3 million to his wife of 14 months.