WASHINGTON—President Donald Trump’s son-in-law and senior White House adviser, Jared Kushner, plans to travel to the Middle East this week to try to advance U.S. efforts to reach an Israeli-Palestinian peace deal, a White House official said Sunday.

The trip marks the White House’s first major follow up to Mr. Trump’s trip to the region last month and suggests Mr. Kushner’s policy portfolio is far from shrinking despite scrutiny by federal investigators into his meetings with Russian officials.

Mr....