The 46-year-old star has moved out of the luxurious London home she shared with the aristocratic art dealer - who is the godson of Camilla, Duchess of Cornwall - after his family became "alarmed" at how close the couple had quickly become.
A source close to Henry told the Daily Express newspaper: "She had moved in within a month of meeting Henry last November and made herself quite at home.
"Most of the family were rather alarmed by how besotted he was and feared she had delusions of landing herself a title - she used to joke about becoming Lady Love."
Courtney - who is the widow of Nirvana frontman Kurt Cobain - recently admitted she was into "really, really rich" guys, as she thought she could be a "real asset" to a wealthy man like future Lord, Henry.
She said: "These days I`m only interested in plutocrats. Like really, really rich guys. I`m determined to land one sooner or later. The thing is, I think I can be a real asset to a wealthy man."
The Hole singer - who has a history of drug addiction - also previously claimed to be an "Anglophile".
Courtney explained: "I am an Anglophile and love England so much. I used to live there and always consider myself half English."