"They are into one another," said a source to the US media. "[He] is very charismatic and she loves that about him. He's a go-getter."
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Demi is said to have first met 60-year-old Peter during the time that she was romantically linked to his 31-year-old son and has already introduced him to her family.
Friends of the Ghost actress have revealed that during the time she spent getting to know Harry's family she became increasingly close to his charismatic father, who cofounded the Hard Rock Café in 1982 and is said to be worth around £180million.
Restaurants run in the Morton family. Peter's father was the famous steakhouse founder Arnie Morton and his son Harry owns two branches of Pink Tacos in Los Angeles.
Peter and Demi reportedly bonded over their mutual experiences of divorce and both have three children.
For 13 years Demi was married to actor Bruce Willis, 58, who is the father to her children, Rumer, 25, Scout, 21, and Tallulah, 19. Five years after her divorce from Bruce in 2000 Demi married That '70s Show actor Ashton Kutcher, who was 27 at the time.
Demi and Ashton went their separate ways in 2011 and are said to be headed for a £190million divorce battle after continuing to disagree about how they should split their considerable wealth.