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When deciding whether to run for office, Vice President Joe Biden has made it a practice to seek his family’s counsel. That advice has included at least two members of his immediate family—his sons—urging him to run for president in 2016, Biden friends and advisers say.

Before his death last month, elder son Beau Biden encouraged his father to get into the race, people familiar with the matter said. And Hunter Biden told a friend in recent weeks he, too, would like to see the vice president wage one more...