Mr. Biden’s increasingly bitter fight with Mr. Trump comes at a potential tipping point in the 2020 Democratic presidential race. Until recently, Mr. Biden had been leading in most national polls since he entered the field last spring. But Senator Elizabeth Warren of Massachusetts, fueled by a highly disciplined campaign, has surged in many early-voting states.

A Monmouth University national poll released Wednesday showed Mr. Biden locked in a statistical dead heat with Ms. Warren, who garnered 28 percent support from Democratic and Democratic-leaning voters to his 25 percent, a difference within the survey’s margin of error.

At Wednesday night’s rally, Mr. Biden took a swipe at some of his Democratic opponents who had accused him of focusing too much on Mr. Trump and not enough on issues facing voters.

“A lot of my opponents say we have to do more than just beat Donald Trump,” he said. “I agree. We have to do more than beat Donald Trump. We have to beat him like a drum.”

Mr. Biden’s team has been alarmed by reports that Mr. Trump’s supporters plan to launch an aggressive advertising campaign to portray the former vice president as the person who had acted improperly — despite the lack of any evidence to support that claim.