TOKYO—Subaru has a problem any auto maker would love to have. It's selling too many cars.

Subaru-brand sales in the U.S. are up 27% in the first seven months of 2013, more than three times the market overall. The auto maker, owned by Fuji Heavy Industries Ltd., is on track for a sixth-straight year of sales gains, a period that includes the 2009 financial crisis that bankrupted two of Detroit's Big Three.

But...