The Beatles legend, who underwent secret heart surgery to increase blood flow to the arteries last autumn, is said to be devastated by Heather's ultimatum and the furious rows have taken a toll on his health.
Heather has reportedly threatened to take four-year-old Beatrice to the US in a bid to force Paul, 65, to agree to a divorce settlement.
One of Paul's friends told Britain's Daily Star newspaper: "Arguing with Heather is the last thing Paul needs."
"To recover from heart surgery you need peace and relaxation. Instead, he is getting even more rows with Heather."
"He is under a lot of stress which can't have helped his heart condition."
Heather - who owes $4 million in unpaid legal bills to her former lawyers - has accused Paul of prolonging their divorce battle until she runs out of money.
The former model also believes Paul is trying to "buy" Beatrice's love after discovering he spent $30,000 on Christmas presents for his daughter.
The musician splashed out on a top-of-the-range Bang and Olufsen sound system, karaoke machine, a Barbie 4x4 car, 10 Barbie dolls, 20 outfits with shoes from exclusive London department store Harrods, a children's electric guitar, keyboard, and various classical and pop CDs.
A source told Britain's Daily Mail newspaper: "She was saying what a terrible role model Barbie was for Bea."
"And she was saying how annoyed she was that Paul appeared to be trying to push her into music when Heather is all about Bea growing up a free spirit who shouldn't be pushed - especially at such a young age - into any career."
"The truth of it, of course, was more likely that Paul just wanted to make it a Christmas to remember for Bea."
Next month, Paul and Heather will attend a five-day High Court hearing in a bid to finally agree on a divorce settlement.
Heather could be awarded more than $60 million following a four-year marriage to the star.
