WASHINGTON—The Trump administration asked the Supreme Court to reinstate a series of new federal executions that had been due to begin next week, but were blocked by two lower courts while inmates pursue legal challenges.

The filing Monday came hours after the U.S. Court of Appeals for the District of Columbia Circuit refused to overrule a federal district judge who last month found that the Justice Department’s plans to execute four men with a lethal injection of pentobarbital likely skirted the Federal Death Penalty Act....