WASHINGTON—Record-setting jobless claims and dire economic forecasts are giving fresh urgency to the debate within the Trump administration and across the country over how rapidly coronavirus-fueled restrictions should be pared back so the economy can begin its revival.

President Trump has expressed eagerness to move quickly, and Treasury Secretary Steven Mnuchin said on CNBC Thursday that he thought the U.S. economy could be ready to reopen by the end of May, “as soon as the president feels comfortable with the medical issues.”

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