Chrysler Group LLC and Nissan Motor Co. said U.S. sales in July rose at double-digit rates, indicating that the auto industry’s recent strong sales would persist.

Chrysler said sales rose 20% in July, as its Jeep brand turned in another month of strong growth. Nissan, meanwhile, said its sales improved 11%, with gains occurring in both its Nissan and luxury Infiniti divisions.

The auto-industry’s sales have continued to fare…