WASHINGTON — It’s the stuff of a campaign manager’s dream: The sitting vice president, in the midst of his own run for president, dashes across Pennsylvania Avenue and bursts into the Senate to cast the deciding vote on make-or-break legislation, saving the day for his party while C-SPAN cameras capture the moment.

For Joe Biden, it could become a reality — in the event of a deadlocked Senate after the midterm elections.

If the Senate splits evenly between Democrats and Republicans, the vice president’s role as the 101st senator would instantly be elevated.

That in turn would raise Biden’s own profile heading into 2016, when Biden has said he may run for president again. It could even help him try to rival the rock-star status that Democrats have already bestowed upon Hillary Rodham Clinton.

“It makes Joe Biden suddenly a hugely relevant Washington figure,” said Douglas Brinkley, a presidential historian at Rice University. “It shows he’s a power player.”

On the other hand, it could also make Biden the face of messy political fights on Capitol Hill, potentially alienating some voters along the way.