"It's somewhat serious," an insider tells People magazine. "He introduced her to a bunch of his friends."
The newly brunette actress, who supposedly avoided A-Rod's onetime pal Madonna at a polo match over the weekend, was also photographed in primo seats at Yankee Stadium on Tuesday night as she watched her squeeze's team trounce the Texas Rangers.
"He was a charmer," a source tells Us Weekly, "working it with Kate."
It wasn't the first time Hudson cheered on the Yankees as they took on the Rangers. She accompanied Rodriguez to Dallas late last month for a three-game series against Texas, where Us says they shared the "rock star" suite at the Hotel ZaZa and People reports they worked up a sweat together in the hotel gym.
Along on the trip was her 5-year-old son, Ryder (dad is her ex-husband, Chris Robinson), who attended one of the games with his nanny while Hudson caught it on TV at the hotel, according to People.
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"She remarked several times about what a big baseball fan she is!" recalls an eyewitness.
But one source wonders if the romance is destined to be a home run, especially given Hudson's dating history, which includes her repeatedly rekindled romance with Owen Wilson and her brief but intense interlude with Lance Armstrong.
"You never know with Kate," the snitch shrugs to the mag.
As for Rodriguez, he's apparently been smitten with the star ever since he caught her Oscar-nominated performance as groupie Penny Lane in "Almost Famous."
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"It's his favorite movie," a confidant tells the National Enquirer. "Alex has seen it dozens of times, and he fell for Kate the first time he saw it."
Hudson and Rodriguez were initially seen together in November, when they engaged in a flirty tête-à-tête at the Fontainebleau Hotel in Miami.
A-Rod, who was linked to Madonna at the time, was apparently all too aware that sparks were flying.
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At one point, as he watched Hudson walk away after shaking her groove thing in his direction, he turned to a pal and remarked, "That's trouble."