The serial dater, who has previously been linked to Jennifer Aniston, Jessica Simpson and Kim Kardashian, is said to be 'crazy', about the Desperate Housewives actress.
And following the breakdown of her marriage to Tony Parker, Mayer is thought to be planning a charm offensive.
A source told The Mirror: 'John is crazy about Eva. There's something about newly divorced ladies that drives him nuts. He was the same over [former partner] Jessica Simpson when she left Nick Lachey and now he has his eye on Eva.'