Odom, 34, is asking for $10million and even demanding the return of her $875,000 engagement ring in exchange for ending his whirlwind romance to the reality star.
The bombshell conditions revealed in the National Enquirer also include keeping his silence about the alleged existence of a sex tape.
The basketball player, who has been battling a serious drug addiction, allegedly wants the money in exchange for keeping quiet about the inner secrets of the Kardashians.
A source told The Enquirer: 'Kris and Khloe are very worried about Lamar... He knows all about their extreme surgery procedures - when, how often and how much - and the pressure put on the younger girls to follow suit.'
The source also claims Odom, who was seen heading for training at his alma mater in Rhode Island on Tuesday, could tarnish Khloe’s career by exposing a sex tape the couple made.
'Lamar is also prepared to destroy Khloe’s reputation by discussing their sex tape, and how she targeted him and tore his family apart,' the source told the magazine.
'Kris is insisting on a confidentiality agreement to ensure Lamar can never discuss his life with the Kardashians either in print or verbally.' The source added Lamar would not sign anything until the money was paid.
The outing came as yet O.J. Simpson's manager spoke out to claim the jailed American Football star could be Khloe's father.
However Pardo's allegations owe more to a hunch than any inside knowledge.
'Whenever I bring it up, he giggles,' he said.
While Kris and her then-husband Robert Kardashian were close to the Simpsons, this is the first time OJ has been named as a potential father to Khloe.
But Pardo claimed: 'They all took their vacations together. There was a lot of partying going on back then. Kris cheated on Robert - she was known for having a good time.'
Kardashian also served as a member of Simpson's 'Dream Team' as his attorney in the infamous 1995 murder trial.
While Kris Jenner has openly admitted cheating on her former husband, she has always strongly denied any suggestion that Robert was not her daughter's father.
'I have never heard such crap in my life,' she told GMA in 2012.
'I was there, right. I gave birth, I know who the dad was. Everything’s good. We’re all good. It’s the family joke because Khloe was blond and not dark-haired.
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