Countries that haven’t yet hit a U.S. limit on imports of Iranian oil can continue the trade without risk of sanctions until they reach the negotiated cap, a senior U.S. official said.

The U.S. in November issued waivers for China, India and six other countries to continue importing limited amounts of oil from Iran. But the waivers expired in early May and the administration said no new waivers would be granted, as the Trump administration pursued a pressure campaign against Iran that has heightened tensions between Washington...