The Sun claims to have learned from an inside source that the England left-back reacted angrily when the club only offered him a one-year extension to his current deal.
Cole is in the final year of his contract and could join another club for free when the transfer window re-opens if the issue remains unresolved.
A club insider told the newspaper: "[Roman] Abramovich has a rule that anyone over 30 can only be given a one-year extension but it's madness in Ashley's case.
"Everyone knows he's the best left-back in the world.
"He trains hard, really looks after himself and can play at this level for many more years to come. There isn't a club in the world who wouldn't want Ashley on a free."