When confronted by Wallace about the disparity between what his public health officials are saying and what he has said about the timing of the vaccine, Trump stuck with his previous comments.

"We can have it a lot sooner," Trump said. "It's a very political thing because people like this would rather make it political than save lives."

Top U.S. government public health officials and other experts have said that it's possible -- if everything goes well, but unlikely -- that vaccine trials will have enough clear and compelling data to be available by the end of October or beginning of November. Dr. Anthony Fauci, the nation's foremost infectious disease expert, has said he thinks there will be a "safe and effective vaccine" by "the end of this calendar year."

Fauci has also said that the United States will not return to "normal" until the middle of next year.

Biden interrupted the president by referring to Trump's previous statements earlier this year, when he said the virus would be gone by Easter and then also referenced Trump's potentially dangerous comment that people can inject bleach to get rid of the virus. Trump then responded that he said the bleach comment "sarcastically" made.

Biden turned to the hundreds of thousands of Americans who have died due to the coronavirus.

"His own head of the CDC said if we just wore masks between now -- if everybody wore masks and social distanced -- between now and January, we probably save up to 100,000 lives," Biden said.

Wallace pivoted the conversation to the global COVID-pandemic, and Biden began by listing statistics including 200,000 Americans dead and over 7 million infected.

"The president has no plan. He hasn't laid out anything. He knew all the way back in February how serious this crisis was. He knew it was a deadly disease," Biden said. "And I laid out again in July what we should be doing."

You should get out of your bunker and get out of the sand trap and your golf course and go in your Oval Office and bring together the Democrats and Republicans and fund what needs to be done now to save lives," he continued.

Trump countered by saying if the country had listened to Biden then millions of people would have died, not 200,000, and touted his decision to restrict travel from China in February, along with his efforts on ventilators and vaccine development.

"When you talk about numbers you don't know how many people died in China, you don't know how many people died in Russia, you don't know how many people died in India. They don't exactly give you a straight count," Trump said.

While on the topic of the Supreme Court and the fate of the Affordable Care Act, Trump was quick to call Biden's health care plan "socialized Biden."

Biden, notably, has said he does not support "Medicare for All" and responded by calling out Trump for promising to repeal and replace Obamacare -- which he hasn't.

"He does not have a plan," Biden said. "The fact is this man doesn't know what he's talking about."

Less than 20 minutes into the debate, both candidates have interrupted each other multiple times, making it difficult to understand what either is saying. Wallace has specifically asked Trump multiple times to let Biden finish his thought.

Eventually, the cross-talks continued for long enough that the former vice president said, "Will you shut up, man?" to the president.

Despite Republican successes in statewide races in Ohio since 2016, the Buckeye State has become increasingly competitive as Trump and former vice president Joe Biden prepare to face off in Cleveland for their first debate on Tuesday.

In recent weeks Trump and Vice President Mike Pence have traveled repeatedly to Ohio, where they defeated Hillary Clinton and Sen. Tim Kaine, D-Va., her vice presidential candidate, in 2016, 51% to 43%.

Republicans also performed well statewide two years ago, winning the gubernatorial race and other down-ballot contests.

“Ohioans showed up for President Trump in 2016 and we expect them to do the same this November,” Trump campaign spokeswoman Samantha Zager told ABC News.

The president’s campaign has had a permanent presence on the ground in Ohio since 2016, and said they have made more than 10 million voter contacts in the state, where they and other GOP groups continue to canvass on the ground during the coronavirus pandemic.

The Biden campaign, meanwhile, has relied on virtual and digital organizing in the state during the pandemic, and has hosted hundreds of events and held 450,000 conversations with voters over the last month. Following Tuesday's debate, Biden will take a train tour of the state on Wednesday.

Biden has delivered a sharp economic message focused on Trump's record, centered on Toledo and Youngstown, home to several car plants and many blue-collar voters who abandoned Democrats and voted for Trump for years ago.

He's run at least eight television ads in recent weeks, including a response to Trump's call for a ban on Goodyear, the Ohio-based tire company than ran in Akron, where the company is based.

In an election year reshaped by the coronavirus pandemic, it’s also playing host to a number of legal battles over access to ballot drop boxes, which are increasingly used by voters as an alternative to in-person voting or relying on the United States Postal Service, amid concerns about service delays.

Read more about the battle for Ohio here.

-ABC News' Ben Siegel and Adia Robinson