Analysts say the mudslinging may be the worst in U.S. history.
"There are aggressive campaigns. There are nasty campaigns," said Julian Zelizer, a presidential historian at Princeton University. "But when the president starts tweeting like this, it's as low as you can go."
Some of Trump's attacks are recycled from 2016, when he questioned Hillary Clinton's health, said she should be imprisoned and demanded she take a drug test before their debates. The relentless attacks helped undermine her campaign.
"This is a campaign from the very beginning that has struggled to define Joe Biden in the way that they did Hillary Clinton," said Amy Walter, a political analyst from Cook Political Report. "It's not working this year."
With Biden leading in the polls, and voters increasingly skeptical of Trump's handling of the COVID-19 pandemic, which has killed more than 195,000 Americans in six months, Trump has sought to convince voters that Biden poses a bigger danger.
"He cannot afford this race to be a referendum on himself," Walter said. "But he cannot help himself."
Insults are trickier for Biden, whose campaign is based in part on his claims to decency and empathy.
Still, he called the president a "climate arsonist" on Monday for failing to address global warming as wildfires rage across California and Oregon. His campaign manager on Tuesday blasted Trump's crowded indoor campaign rallies in a pandemic, saying "people will die because of these types of events."
Sometimes Biden's most alarming claims have proved accurate. He drew sharp criticism in April, for example, by speculating that Trump would try to delay the Nov. 3 election. Three months later, Trump suggested doing exactly that, although he has no authority to order a delay.
A spokesman for Trump's campaign did not respond to questions Tuesday.