Michael Kors Holdings Ltd. is close to a deal to buy Italian fashion house Gianni Versace SpA for about €2 billion ($2.35 billion), people familiar with the matter said Monday, in a move that would put one of the glitziest names in high fashion in the hands of a budding U.S. conglomerate better known for affordable luxury.

A deal would give Michael Kors a sought-after foothold in European fashion. Although Mr. Kors, who is the chief creative director of his company, has worked in high-end fashion, his brand is now best known...