WASHINGTON—GOP Rep. Darrell Issa of California has won eight elections to the House. Republican presidential nominee Donald Trump could make his ninth a little tighter.

Ousting Mr. Issa, the wealthiest member of Congress and a powerful former committee chairman, from his district north of San Diego would be an unexpected and welcome upset for Democrats in November. But Mr. Trump’s inflammatory campaign style and slipping poll numbers has Democrats believing they have an opening, particularly after the party’s nominee for the...