Fiat Chrysler Automobiles NV and Renault SA are in talks for a wide-ranging tie-up that could include joining large portions of their businesses, according to people familiar with the matter, a move that could reshape the European auto industry.

While many details and the financial contours of any transaction couldn’t be learned, it could be substantial: Fiat Chrysler, which trades in New York and Milan, has a market value of $20 billion. Paris-listed Renault is worth about $17 billion.

There...