"We're friends," Rodriguez, 33, says.
Asked if there is any romance between them at all, Rodriguez reiterates, "We're friends – that's it."
While rumors have swirled that the slugger and the singer, 50, have been carrying on a romantic relationship and flying around the country in order to spend time together, "I can tell you this," Rodriguez says, "I have never been on a plane with her."
He also finds much of what's been written about them laughable: "I've been to two [of her] concerts, yet I've read that I went to 20. I've also read that we were buying an apartment together. That is absolutely ridiculous and not true."
"You have to have a sense of humor when it comes to this stuff," Rodriguez adds.
When pressed on how much time they have spent together, and whether he had ever stayed the night at Madonna's home, Rodriguez laughed: "If I answered every rumor we'd be here for three weeks."
(A source close to Rodriguez tells PEOPLE that reports of his spending the night with Madonna are "completely, unequivocably false.")
Focused on CharityWhile Rodriguez and Madonna were both in Mexico City last weekend, Rodriguez says his time was focused on running a baseball clinic for kids.
"When you're among 200 kids and teaching the game of baseball it gives you an amazing feeling," he says, "just that you may inspire kids to reach their goals, whether it's baseball or whatever their goals might be."
The desire to help children is something he and Madonna have in common – and one reason he has become a supporter of Madonna's Raising Malawi charity.
"I initially knew nothing about her charity," Rodriguez says. But after being invited to one of Madonna's fundraisers, he found that "we shared a lot of the same commitments toward helping children. I was very moved by the whole thing. I saw the documentary and it was very inspiring and informative. I saw an opportunity to help and I jumped at that."
Rodriguez jumped into three other philanthropic efforts on his own recently as well: He's funding the completion of a Pediatric Outpatient Unit at Bronx-Lebanon Hospital Center in New York City; funding a Community DentCare Network at Columbia University College of Dental Medicine (to provide preventative dental services for kids); and completing a new learning center at the Boys and Girls Club in Miami that A-Rod himself frequented as a child.
"If it wasn't for the Boys and Girls Club, I wouldn't be where I am today," Rodriguez says. "So I feel a great responsibility to not only give money, but to be there, and to be seen by the kids. I still hang out at the Boys and Girls Club."
Dealing With His DivorceRodriguez, who amicably settled his divorce with wife Cynthia, 35, in September, also addressed rumors that he and Cynthia had been arguing.
"I have a wonderful relationship with Cynthia," says Rodriguez, who shared Thanksgiving dinner with his ex and their two children, Natasha, 4, and Ella, 7 months, last week. "We try to be the best parents we can."
As for the amount of time he spends with his children, "I see my children, honestly, seven days a week, which has been great. We both have a lot of respect for each other and she's a great lady."
As for Madonna: "I have a lot of respect for her," he says. "She's very committed to making the world a better place. She's had an amazing career, and we're friends. That's it."
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