The actress, who split from her director husband Sam Mendes in March, wants to keep the relationship out of the public eye, according to the News of the World.
A source said: "Louis has been spending a lot of time with Kate recently and they were in New York together while he was shooting an advert. They've been on lots of dates and get on really well. It's early days for them but they enjoy each other's company.
"They have been desperate to keep the relationship under wraps so they've been a bit low-key so far. Louis has a good sense of humour and really is a lovely guy.
"He loves to surf and he's really quite engaging and very down-to-earth. He's a good looking guy and gets lots of attention when he's out and about - but he's not your typical model. Not at all big-headed.
"They are trying to plan their diaries to make sure they can spend as much time together as they can. They hope to visit Cornwall together in the summer for a holiday."