US Vice President Mike Pence and members of the CONVID-19 task force speak to the press at the White House March 10, 2020, in Washington, DC.

There was also discussion of making the payroll tax rollback permanent, an official, who declined to be named, told CNBC earlier Tuesday . Payroll taxes are used to fund Medicare and Social Security.

The president "unveiled his proposals, strong proposals, for a temporary payroll tax cut holiday, which I think he would prefer to last through the end of the year," Kudlow said.

That proposed tax change would last through the rest of the year, and would apply to both employers and employees, a White House official told CNBC. Kudlow confirmed those details at the briefing Tuesday evening.

Kudlow's comments at a White House briefing on the virus came hours after Trump pitched implementing a 0% payroll tax rate during a closed-door meeting with Republican lawmakers on Capitol Hill.

Larry Kudlow, President Donald Trump's top economic advisor, offered few details Tuesday when reporters pressed him for specifics about the administration's plan to offer economic relief from the impact of the new coronavirus through a "temporary payroll tax holiday."

The White House is also considering federal assistance for the shale industry, CNBC reported, as oil prices have recently plummeted due to a price war between Russia and Saudi Arabia.

At the briefing, Kudlow suggested that growth resulting from the payroll tax deduction could make up for the potentially enormous amount of money lost.

Asked how that could be done, Kudlow said, "Let us put the proposal out in concrete details and flesh that out and we'll have much better answers."

CNBC's Eamon Javers had pointed out to Kudlow that federal payroll taxes generated $1.17 trillion in fiscal year 2018, according to the Center on Budget and Policy Priorities.

"Right now I want to stay in my lane, and I think the health story, the coronavirus story, is very, very important here," Kudlow said. "We will do the best we can ... to give you specific plans and details once we flesh them out."

Kudlow was joined in the White House briefing room by other members of the coronavirus task force, led by Vice President Mike Pence, who fielded questions on the administration's response to the fast-spreading virus.

Pence assured that the risk to the average American of contracting the disease "remains low." But, Pence added, "for senior citizens with serious underlying, chronic health conditions, the potential for serious consequences is very real."

Reporters then asked how lifting the payroll tax — a potentially massive deduction, especially if applied to both employers and employees — could be done without knocking the budget off balance.

"The payroll tax holiday is a bold move. It's a very bold move and this is a bold president," Kudlow responded.

"I think there will be a big growth payoff. I think it'll help deal with whatever challenges occur in the next few months."

The payroll tax cut is intended for "ameliorating the tax burden on the middle class," Kudlow added. "That's really what the payroll tax is principally about."

"I think we're going to get a big growth kicker," he said. And while the labor market "may still a bit or not," Kudlow predicted that "I think over time we'll make it up with much better economic growth."

Markets have fluctuated wildly amid the spread of the virus. Stocks suffered through a historic sell-off Monday, but clawed back much of those losses during Tuesday's trading session.

The coronavirus, which is believed to have originated in the Chinese city of Wuhan, has quickly spread to more than 118,000 cases worldwide and has killed more than 4,200 people, according to data compiled by Johns Hopkins University.

The Trump administration and lawmakers at every level are working to contain the spread of the virus in the United States. At least 805 cases are currently confirmed in the U.S., and at least 28 people have died, according to Johns Hopkins.

-- CNBC's Eamon Javers contributed to this report.