When John Idol met Donatella Versace for dinner earlier this year at a private home in Milan, Ms. Versace told the Michael Kors Holdings Ltd. boss that the Gianni Versace SpA fashion house wasn’t for sale.

On Tuesday, Michael Kors said it had agreed to pay $2.12 billion for the Italian luxury brand founded by Gianni Versace, furthering the U.S. company’s ambition of becoming a global luxury fashion group.

“I wasn’t thinking about selling,” said Ms. Versace, who took over the business after her brother’s death in 1997. “But I talked to John. He has the same passion for the business. And he can offer us expertise” in digital commerce, accessories and growing the store base, she said in an interview.

The deal marks one of the first attempts by an American fashion company to run an elite European brand. Once it closes, Michael Kors will change its name to Capri Holdings Ltd. and trade under the symbol CPRI. In addition to its namesake handbags, the company owns high-end shoemaker Jimmy Choo, which it acquired for $1.2 billion in 2017.

Ms. Versace will remain Versace’s chief creative officer, and members of the family, including her brother Santo and daughter Allegra, will receive €150 million ($176 million) of the purchase price in shares of Capri Holdings. Versace’s management team will continue to be led by Chief Executive Jonathan Akeroyd.