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The 38-year-old actress has booked a secluded luxury cottage in America's East Hamptons resort so she and the 65-year-old Beatles legend can get to know each other better.
A source said: "It's early days but Renée really thinks they could have something.
"The Hamptons seems to be a lucky place for them. So Renée is playing it safe by spending the weekend with him there.
"The cottage has a piano and guitars, so she might be serenaded with an impromptu performance."
Renée and Paul first met at a Tom Petty concert in New York last month where they were seen cosying up to one another.
For their second date the Oscar-winning actress and the beloved musician enjoyed a romantic dinner at The American Hotel restaurant in The Hamptons. The pair met up again at Jon Bon Jovi's East Hampton holiday home last week.
Paul is currently locked in a bitter divorce battle with Heather Mills - mother of his three-year-old daughter Beatrice - and he has recently been spotted with supermodel Elle Macpherson and former model Christie Brinkley.
But despite the amount of female attention Paul has been receiving of late, he and Renée are said to be smitten with each other.
The source added: "'Jerry Maguire' is one of his favourite films, and she is a huge Beatles fan. So they were off to a good start before they even met."