President Donald Trump on Monday will call for new steps to combat the opioid epidemic, including a push to reduce opioid prescriptions by a third over three years, asking the Justice Department to seek more death-penalty cases against drug traffickers under current law, and for federal support to expand the availability of overdose-reversal medication.

The proposals will come in a speech in the hard-hit state of New Hampshire. They form part of a broader blueprint by Mr. Trump, which senior White House officials on Sunday described...