Women's Wear Daily, which is regarded as the industry bible, said they had been told that Chiuri has finalised her contract.
Both Dior and the Italian brand Valentino, where Chiuri has worked alongside her longtime creative partner Pierpaolo Piccioli since 2008, refused to comment.
Under the pair's leadership, Valentino has become one of the hottest and most profitable designer brands in Europe.
If her appointment is confirmed, she would be following not only in the footsteps of the house's legendary founder Christian Dior but also of such designers as Yves Saint Laurent, Marc Bohan, Gianfranco Ferre and John Galliano.
Dior has been without an artistic director since the shock departure of the Belgian Raf Simons in October, which sparked soul-searching in the industry about the pressure creators were now under.
Rumours have been linking Chiuri -- known for her romantic, intricate style -- and high-end label, founded in 1946, for some time.
Dior's chief executive Sidney Toledano told AFP last month that they would "soon say" who was to take over.
"We turn the pages at our rhythm," Toledano said, insisting they would not be hurried into making a decision.
"There are extremely important issues at stake about the direction (Dior) will take in the years to come," he added.
The duo of Swiss designers Lucie Meier and Serge Ruffieux, who have been holding the fort since Simon's exit, are likely to move on to Dior's haute couture range.
Toledano said they were in charge of Dior's forthcoming couture show in Paris on 4 July.
Even before they took over at Valentino, Chiuri and Piccioli had forged a tight creative partnership over two decades, with her hiring him to join her when she was snapped up by Fendi in 1989.
Dior Couture has seen its growth fall by four percent in the last quarter, with its turnover also down by one percent to 1.39 billion euros ($1.5 billion).
The French luxury goods giant LVMH, owned by Bernard Arnault, France's richest man, owns 70.8 percent of Dior.
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