WASHINGTON—President Donald Trump is crafting executive orders that would institute sweeping changes to U.S. refugee and immigration policies, including a ban on people from countries in the Middle East and North Africa deemed by the new administration as a terror risk, according to people familiar with the plans.

A separate order also would lay the groundwork for an escalation of U.S. military involvement in Syria by directing the Pentagon and the State Department to craft a plan to create safe zones for civilians fleeing...