Democrats worried about Hillary Clinton’s electoral weaknesses see Vice President Joe Biden as a potential solution—as long as the two former cabinet colleagues and sometime rivals can smooth their complicated relationship.

Mrs. Clinton’s vulnerabilities were apparent over the weekend when she suffered lopsided losses to rival Sen. Bernie Sanders in Democratic caucuses in Washington, Hawaii and Alaska. The trio of states contain either many white voters or veer to the left wing of the Democratic Party, two constituencies where Mrs. Clinton has struggled.

Mr. Biden, who often talks about his upbringing in Scranton, Pa., in a family that endured financial hardships, has shown an affinity with working-class whites that could help overcome doubts about Mrs. Clinton’s candidacy. He appears willing to hit the trail for the potential Democratic ticket.

Looming over the possible collaboration, however, are tensions between two of the nation’s most important Democrats: a sitting vice president and a former secretary of state who might wind up with the job he has long coveted.

In one notable instance, Mr. Biden said in a January television interview that Mrs. Clinton was “relatively new” to the issue of income inequality and that no one doubted Mr. Sanders’s “authenticity” on that issue.