20 per cent of the brand, known for its Medusa head logo, was bought by United States private equity group Blackstone in 2014, and the family owns the rest.
London-based Kors a year ago snapped up shoemaker Jimmy Choo Plc for about 896 million pounds ($1.2 billion), expanding beyond its namesake brand.
Others have expressed interest in buying Versace, including Paris-based Kering, but some are insisting the price is too high.
Michael Kors, the fashion retailer known for upmarket but affordable apparel and accessories, is to pay between $2 billion and $2.4 billion for Versace, closing a chapter in the colourful story of one of the world's best-known brands.
Michael Kors could not be reached for comment. "I believe he is going to create some sort of brand diffusion with the essence of Versace's DNA".
USA fashion group Michael Kors Holdings Ltd has sealed an agreement to take control of Italy's Gianni Versace SpA, three sources familiar with the matter said on Monday, adding that an official announcement would be made this week. And it comes as the Michael Kors business still hasn't been turned around, despite some progress made.
London-based Kors last year snapped up shoemaker Jimmy Choo Plc for about $1.2 billion
The deal comes as budding luxury conglomerates, including Michael Kors' U.S. rival Tapestry, owner of Coach and Kate Spade, are trying to make inroads into an industry still dominated by major European players, including Louis Vuitton owner LVMH.
Versace and Blackstone both declined to comment to Reuters.
Versace reported sales of €686m in 2016.
At the time of founder Gianni Versace's death, the company had been making plans for a listing on the New York Stock Exchange, but it has remained private ever since. Donatella Versace holds 20 percent, her brother Santo 30 percent and Donatella's daughter Allegra 50 percent.
Last year it returned to a net profit of just under 15 million euros compared with a net loss of 7.9 million euros the previous year. In a tribute to Gianni, Donatella Versace brought back the most iconic looks from his collections spanning the 1990s, including silk blouses mixing leopard prints with images of baroque architecture.