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Looks like Kate Hudson and Alex Rodriguez have officially moved past the summer fling stage of their relationship.

On the heels of reports that the actress has made herself at home in the New York Yankees slugger's Manhattan digs, the flames of four-plus months set off a house-hunting expedition a few days back, reportedly checking out high-end digs (as in $10 million and up) in Pacific Palisades and Malibu, where Hudson already owns a $10 million beachfront pad.

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Joining them on the real estate recon were A-Rod's two daughters (with ex-wife Cynthia), 4-year-old Natasha and 1-year-old Ella, both of whom appeared quite comfortable with Kate, with the former snapped holding her hand and the latter hitching a ride in her arms. Hudson's 5-year-old son Ryder (with ex-hubby Chris Robinson) appeared to be a no-show.

The outing not only sparked speculation that the pair is taking a serious leap forward in their relationship by making a major real estate commitment, but it also touched off baby talk thanks to Kate's wardrobe choice.

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The ostensible stork evidence, as cited by the London Sun (and by "evidence," we mean completely spurious conclusion-jumping with punny "You hiding tum-thing, Kate?" headline): The kicky red minidress Kate was styling while checking out the luxury pads seemed to bunch a bit in the middle.

Last month, In Touch alleged that Hudson, who faced similar belly scrutiny in early 2008 because of some roomy ensembles, was itching to procreate with Rodriguez.

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"She just turned 30 and she's ready to have her second child," claimed a snitch. "She wants Ryder to have a sibling, so she brought up the idea to Alex and told him that they would make a beautiful baby together, and that she would assume all financial responsibility."

But when we checked with Kate's rep about the pregnancy rumblings, he was quick to dismiss the reports, telling Wonderwall succinctly, "She is not pregnant."