WASHINGTON—The U.S. House will try this week to vote on a Republican-written immigration overhaul, its second attempt in as many weeks, as GOP lawmakers balance pressure from President Donald Trump and the party’s base demanding a harder line on illegal immigration against a broader public outraged over the forced division of families under the recent administration crackdown.

Even if congressional leaders manage to bring legislation to a vote, members say it is all but certain to fail, underscoring the party’s lack of consensus...