Alex Rodriguez and his fitness instructor girlfriend Jaclyn Cordeiro have split … TMZ has learned.

Jac tells TMZ … “To clarify, Alex and I are currently taking some time apart but there continues to be mutual love, care, and deep respect between us and for our families. We remain supportive of one another, and there is absolutely no animosity, drama, or negative story here.”

She adds … “I’m also navigating a serious health matter involving a family member at this time, so my focus is understandably there. Alex and I continue to be very close, and out of respect for everyone involved especially our daughters and loved ones I’d kindly ask that any reporting reflects the compassion and privacy this situation deserves.”

A source close to A-Rod tells TMZ … they’ve been apart for about 2 months and there is nothing but mutual respect between them.