Penney is seeking a replacement for Ron Johnson, who was formerly in charge of retail at Apple. He left the company to run J.C. Penney in 2011, but was pushed out 17 months later.
Ahrendts accepted the position months ago, but will officially start as the senior vice president of retail and online sales next year.
According to the WSJ report, Ahrendts' hiring "was locked and loaded" this summer, but the announcement was delayed as Burberry weighed a successor, a source familiar with the matter said.
Apple is reportedly amplifying the role to include responsibility for online sales.
Ahrendts is the second luxury retail executive this year to announce a departure for the tech giant. Paul Deneve gave up his position as chief of Kering SA 's Yves Saint Laurent brand to take on the role as VP of special projects at Apple.