Bernie Sanders’s supporters want him to stay in the presidential contest, despite an unlikely path to the Democratic nomination after a string of primary losses in March.

The Vermont senator would have to completely upend the race by capturing more than 60% of the remaining delegates to overtake former Vice President Joe Biden and earn the party’s nomination to face President Trump in the November election. Mr. Biden’s allies have called for unity, but Mr. Sanders has shown no sign of leaving the race and has deflected questions...