Washington (CNN) Treasury Secretary Steven Mnuchin on Tuesday pitched Republican senators on a massive economic response package amid the coronavirus crisis with a price tag that is now at a staggeringly high $1 trillion, which would include a first wave of checks to Americans that would cost $250 billion, a source familiar tells CNN.

The eye-popping number underscores the growing fears in the White House and on Capitol Hill that the coronavirus outbreak could send the economy into a damaging recession -- and that aggressive intervention by Washington is needed immediately to reverse course.

The proposal comes on the heels of two major pieces of legislation that lawmakers have pushed to address the growing crisis -- and all told amounts to the most far-reaching economic rescue packages since the financial crisis in 2008.

Treasury Secretary Steve Mnuchin confirmed the figure, saying that at the President's request they've put a proposal on the table that would "inject a trillion dollars into the economy," following the GOP weekly policy lunch on Tuesday.

"It is a big number," he said. "This is a very unique situation in this economy." He said it would be a combination of loans, direct checks to individuals and creating liquidity for small businesses.

Senate Majority Leader Mitch McConnell on Tuesday vowed to keep the Senate in session to approve an earlier relief measure passed by the House and then tackle the next economic stimulus measure, which he described as "phase three."

Asked about concern among Republican members over the House bill, the Kentucky Republican said, "A number of my members think there are considerable shortcomings in the House bill. My counsel to them is to gag and vote for it anyway," he said.

After that, "the Senate will not leave until we have passed yet another bill," he said.

Echoing what the administration is now pushing for, McConnell said "we are examining policy tools to put money directly and quickly into the hands of American families."

Timeline for a third coronavirus relief measure in the Senate

McConnell has tasked groups of senators to come together on an economic stimulus plan in line with what Mnuchin is proposing, according to GOP sources. The goal: Get Republicans on the same page, then negotiate a bipartisan deal with Democrats.

Senate Republicans are trying to see if they can reach an agreement within their conference as early as Wednesday.

McConnell alluded to this at his news conference earlier in the day.

"I've divided -- created -- three task forces among Senate Republicans, each of them tasked with coming up with what could best be described as the next bill," McConnell said. "We're trying to reach an agreement among ourselves as to what Senate Republicans and the administration favor doing next."

Senate Republicans will work to reach an agreement with the administration over that next bill, McConnell said at his news conference, and then will "sit down with our Democratic counterparts and see what we can agree to."

Cap under consideration for stimulus checks

Sources told CNN that the Trump administration is looking to set a cap on individuals eligible for the cash benefit that could go directly to Americans.

What the threshold would be is still under consideration, but likely individuals would need to make below $100,000. It's possible the benefit could be capped for anyone making more than $75,000.

Some Republican senators are cautiously expressing support for a push by the Trump administration for the massive, multi-billion dollar economic stimulus as well as the possibility of sending money directly to Americans.

"I understand that Secretary Mnuchin and the President said that their preference now is toward giving direct assistance to families. I think that's exactly the right thing to do," Republican Sen. Josh Hawley of Missouri told reporters on Tuesday.

The remarks came after Mnuchin said at a briefing at the White House on Tuesday that the administration is "looking at sending checks to Americans immediately."

Indiana Republican Sen. Todd Young told reporters that he is "open" to the idea of direct payments to families of around $1,000. Republican Senator Mitt Romney of Utah has proposed giving that amount of money to every American adult in response to the coronavirus crisis.

"What I'm most focused on is what is most helpful to the individual American worker who is unable to work. What is most simulative to our economy, that is what will lead to greater circulation of money throughout our economy, what is the marginal propensity to consume, as an economist would say, of that individual person receiving that thousand dollars, so where are we going to get the most bang for our buck?" he asked.

In an indication, however, that the Trump administration's proposal is likely to run into resistance, Senate Minority Leader Chuck Schumer plans to contrast his own proposal for coronavirus response with the Trump administration's plan during a conference call with Democratic senators at lunch, according to a Democratic leadership aide.

In a PowerPoint presentation, Schumer will outline his estimated $750 billion plan and "explain the contrast to the GOP's expected proposals of industry bailout and tax cuts," the aide said.

The competing plans highlight the challenge ahead for Republican and Democratic lawmakers in Congress as they grapple with what and how much they can do to provide relief for the American public as the coronavirus outbreak hits the United States with an increasingly dire impact.

Mnuchin said Monday as he left a meeting with GOP senators that he would talk to the Republicans on Tuesday about passing a "general" stimulus package that will be a "big number" but would not say what that figure is.

"We have a lot more work to do," he said, "and we have to do it quickly."

He described the actions taken by the administration and Congress as "business interruption insurance."

White House signals sending direct payments to Americans

CNN asked President Donald Trump and Mnuchin about the logistics of an economic stimulus idea that could give $1,000 checks to Americans, which is gaining some bipartisan support.

Mnuchin expressed some support for the idea and indicated it would be discussed during his Capitol Hill meetings.

"I think it's clear we don't need to send people who make a million dollars a year checks. But we like -- that's one of the ideas we like. We're going to preview that today and then we'll be talking about details afterwards," Mnuchin said.

Trump chimed in, saying, "I think we're going to do something that gets money to them as quickly as possible. That may not be an accurate way of doing it because obviously some people shouldn't be getting checks for $1,000. But we'll have a pretty good idea by the end of the day what we're going to be doing."

Senate Minority Whip Dick Durbin, an Illinois Democrat, said that the $850 billion package Mnuchin is proposing is just the "earliest stage" of the negotiations over the next economic stimulus package.

"We have a list, the administration has a list -- that's how you start the process," Durbin said of the package. "This is just the earliest stage of it."

Durbin also said, "It's way too early to project a number" on how much the airlines would need.

Schumer also held a conference call with his Senate Democratic leadership team Tuesday morning to lay out his own $750 billion plan -- and Democrats suggested even more additions, according to Durbin.

On Monday, Sen. Joni Ernst, an Iowa Republican up for reelection, seemed skeptical about a bailout package for the airline industry. "I think we will have some debate over that," she said.

"I'm worried about a bailout but we want to make sure that we are supporting industry. But I think we need to focus on the American worker right now rather than some of the large corporations," Ernst said.

This story has been updated with additional developments Tuesday.