In June 2006, Rodriguez allegedly became a regular of one of Davis' agencies soon after meeting her a gym, two former call girls told the New York Daily News.
A-Rod, then-married, had asked Davis "what she was doing" the first time they met, the anonymous sources said.
Davis, who has claimed Eliot Spitzer as a client, was about to go out with a boyfriend. She gave Rodriguez a card that he later used to book a two-hour "date" with a prostitute, The News reported.
But Rodriguez really pined for Davis and would court her with flowers and jewelery, the sources said. On one email he wrote that he was "addicted" to her, they said.
Eventually, Davis relented and the two briefly dated -- free of charge, according to the sources.
Davis eventually served three months in jail for pimping and is now on probation. She has written a book about her experience leading a prostitution ring but did not name names for the most part.
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When confronted with supposed copies of steamy email exchanges between her and Rodriguez, Davis did not dispute their authenticity.
"Throughout the years, there were a number of clients that I befriended and it was not uncommon for them to want the women they can't have whether it be the phone bookers or the madam," she told The News. "In regard to Alex, all I can say is our paths have crossed personally and professionally."
She added: "Other people have access to my client records as well as my personal information and I can't control what has been released."
Rodriguez's spokesman did not comment on the report.