A senior White House official said on Sunday that foreign nationals with legal permanent residence in the U.S. wouldn’t be affected by Friday’s executive order restricting entry of visa holders and refugees from seven Muslim-majority countries.

White House Chief of Staff Reince Priebus said green-card holders from the countries designated in the order could face extra scrutiny at U.S. points of entry under immigration officials’ existing authority, but said the order “doesn’t affect them.” Green-card holders are foreign nationals...