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"It's somewhat serious," the source tells PEOPLE in its upcoming issue. "He introduced her to a bunch of his friends."
For her part, Hudson followed Rodriguez on the road late last month to Dallas, when the Yankees played the Texas Rangers in a three-game series. The actress and son Ryder, 5 (dad is Chris Robinson), stayed at the same hotel as Rodriguez, and the couple "worked out together in the fitness center," says an eyewitness.
For one of the games, she sent Ryder and his nanny to the ballpark while she "stayed behind and watched the game from the hotel," adds the eyewitness. "She remarked several times about what a big baseball fan she is!"
Will their relationship last past the season? Another source says it's too early to tell: "You never know with Kate."
• With reporting by DARLA ATLAS and K.C. BAKER
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Note that this week’s cover of PEOPLE may differ regionally
