Former Vice President Joseph R. Biden Jr. is accustomed to fielding questions from voters about picking a running mate. But on Monday, he was asked a particularly provocative one: Would he consider choosing a Republican as his vice president?

“The answer is I would, but I can’t think of one now,” Mr. Biden said at a town hall-style event in Exeter, N.H., drawing laughter from the crowd.

Mr. Biden then elaborated on what he meant. “There are some really decent Republicans that are out there still, but here’s the problem right now of the well-known ones: They’ve got to step up,” he said.