J.P. Morgan Chase & Co.’s Sapphire Reserve credit card has been popular with consumers—maybe too popular.

Some at J.P. Morgan are raising concerns that Sapphire Reserve won’t make money for the bank, due in part to high demand and the card’s generous rewards. Meanwhile, the lender is pushing for about $200 million in fresh cost cuts in the unit that oversees the card, people familiar with the matter say.

The...