Britain’s parliament will vote on Tuesday on a possible inquiry into Prime Minister Keir Starmer, looking at whether he misled the House of Commons ‌over the appointment of former US ambassador Peter Mandelson.

Any such inquiry could have serious implications for Starmer’s future. He has so far resisted pressure to quit over his decision to hire Mandelson, but if found to have knowingly misled parliament his position would likely become untenable.

House of Commons Speaker Lindsay Hoyle said he...