"Tiger and Lindsey have been friends for a while, and nothing started out romantically at all," a source tells PEOPLE. "But they really have a lot in common and got closer and closer. He still refers to her as 'my very good friend,' but he's been spending more and more time talking to her – and talking about her."
Last month, Vonn's reps kept mum about the rumored relationship, telling PEOPLE that her "focus is solely on competing and on defending her titles and thus she will not participate in any speculation surrounding her personal life at this time."
But the source close to Woods tells PEOPLE that Woods, 37, and Vonn. 28, talk and text frequently.
"Tiger really does want a woman who he can have good conversations with," he says. "He wants shared interests and outlooks. He is finding that with [Lindsey]."
And perhaps most importantly…
"She's not freaked out by his past, and that's really appealing to him," says the source.