The Beatles legend, who split from the former model in May 2006 following a four-year marriage, bought the Beverly Hills mansion from rocker Courtney Love as an engagement present for Heather in 2001.
Heather, 39, insists the house belongs to her, but Paul has become emotionally attached to the property and is reluctant to let it go.
A source said: "Neither side will budge. It is deadlock. Heather helped choose the house, she persuaded Paul to buy it and it was an engagement gift to her.”
“That is certainly what she is telling people. To Heather the home represents the glitzy lifestyle she was thrilled to marry into.”
“And she intends to spend a lot of time in Los Angeles, building up her profile over there."
Paul, 65, is unwilling to relinquish the house - situated on Heather Road - because of its "mysterious" ties to fellow Beatle George Harrison's death in November 2001.
The source added to a British newspaper: "Paul has never been a fan of California - he prefers the quiet life. But Heather Road became very important to him around the time of George's death.”
“There is still a mystery, known only to Paul, George's family and a few others about his final hours. His death certificate had a false address.”
“It has been said he passed away in Heather Road in the presence of his wife Olivia and Hare Krishna mentors. Paul invited George to stay at the house just days before he died."
Paul spends most of his time at his $24 million Peasmarsh estate in West Sussex.
He also owns a $4 million Mull of Kintyre estate, in Scotland, a $10 million London mansion and a ranch in the US.
Last week, it was reported Heather had demanded a $100 million divorce settlement from Paul and vowed to take him to court if he didn't pay up.
Heather was even said to be threatening to take their three-year-old daughter Beatrice to the US for an extended stay in a bid to convince Paul to agree to the staggering sum.
