Paul's estranged wife Heather fears that if Paul wipes the only copies of the recordings, they will hinder her chances of a big divorce settlement.
Heather, 38, has told her lawyers to demand Paul keeps the 15 tapes safe so she can use them in court.
A source told Britain's Daily Mirror newspaper: "Heather feels Linda's tapes will back up her claims of suffering abuse by Paul."
"Heather is worried that if Paul has the only copies, they will be destroyed or kept secret, and her case will be seriously weakened."
"Heather believes her story will stun the world. If Linda's tapes underline her case then so much the better. The voice of Linda allegedly disparaging Paul from beyond the grave is potentially dynamite."
Linda, who died of cancer in 1998, made the tapes in the 80s. The recordings, which are described as "emotional" and "cathartic," reportedly see Linda venting feelings she never shared with her closest friends.
The tapes are now owned by Paul Cox - with who Linda wrote a vegetarian cookbook - who is a sworn enemy of the Beatles legend.
Paul, 64, is demanding a gagging order to ensure the content of the tapes is never revealed.
Heather's lawyers Mischon de Reya wrote to her estranged husband's legal firm Sheridans after learning Cox and McCartney plan to meet to discuss the tapes.
The former model's friends have insisted Heather doesn't want to bankrupt her husband and would be happy with a "sensible" figure.
A friend said: "She has been painted as a gold-digger, but this is far from the truth. She's not greedy and would happily settle for a sensible figure to finally put closure to the marriage."
