Press, 62, was the only member of Chrysler's senior leadership team to remain with the company after a fast-track bankruptcy under the supervision of the U.S. Treasury.
A former senior executive at Toyota Motor Corp <7203.T>, Press joined Chrysler in 2007 shortly after it was acquired by its former owner Cerberus Capital Management LP.
A Chrysler spokesman could not be immediately reached for comment. Press was also not immediately available.
Press's departure, which the newspaper said could come by the end of November, would mark the end of a rough ride for a veteran executive who built a reputation at Toyota as a strong advocate for dealers.
Press said he wanted to restore an American icon when he joined Chrysler, but his term at the No. 3 U.S. automaker corresponded with a slide into bankruptcy and a controversial decision to slash Chrysler dealerships.
Italy's Fiat SpA acquired operational control of Chrysler when it emerged from bankruptcy in June.
Fiat Chief Executive Sergio Marchionne has shaken up the company's ranks of operational executives, but has kept a low profile during his more frequent visits to the automaker's headquarters outside Detroit.