(CNN) Senior Trump administration officials urgently pressed GOP senators to approve a House-passed bill to deal with the coronavirus pandemic -- and quickly act on a major economic stimulus package amid growing fears that the outbreak could send the country into recession.

In a closed-door meeting Monday night, Treasury Secretary Steven Mnuchin urged GOP senators to get behind the House bill, which had prompted growing Republican concerns in recent days, with top officials warning the economic situation is too precarious and the threat to public health too severe to let the measure hang in limbo for days.

The cost of the new package will be roughly $850 billion, a source briefed on the matter tells CNN, and will include aid to small businesses and airlines.

Mnuchin signaled Tuesday that after conferring further with GOP senators, the Trump administration is looking at sending checks to Americans rather than a payroll tax holiday.

Asked what the Senate would do when it officially receives the House bill, Senate Majority Leader Mitch McConnell told CNN bluntly: "Pass it."

In the meeting, Mnuchin told GOP senators he would present more details during a Tuesday lunch about the Trump administration's proposal for the next economic package. He also told senators to get behind the House-passed measure, which would ensure individuals have access to free testing for the disease and displaced workers have access to paid leave, bolster unemployment insurance benefits, expand food stamps and increase federal funding for Medicaid programs.

The warning, GOP senators suggested: Act now before it's too late.

"I don't think we can assume we can keep reconvening the Senate every week, like we did this week," said Sen. Marco Rubio, a Florida Republican, after the meeting. "I think the assumption is going to be, we're going to do something (and) it's going to be big because we can't assume we can just keep coming back here."

At the meeting Monday, Mnuchin did not detail a price tag for the next economic stimulus measure, but earlier in the day, Senate Democratic leader Chuck Schumer of New York proposed $750 billion in the next plan. Republicans did not reject that number out of hand and said that Congress would likely have to act on a subsequent stimulus measure once the next plan is approved. Already, Congress has appropriated $8.3 billion to deal with the crisis.

"We have a real focus on urgent action," Eric Ueland, White House legislative director, told reporters after the meeting. "I'm hoping there can be swift action on consensus items."

Ueland and Mnuchin told reporters that the House bill, which the treasury secretary negotiated with House Speaker Nancy Pelosi, should be passed immediately despite the concerns among some Republicans that the measure could hurt small businesses.

"Again we'll be speaking to the Republicans at lunch tomorrow about the bill but we look forward to them taking it up and passing it," Mnuchin told reporters. "But we got a lot more work to do and the main reason why we're here tonight was talking to the Senate about other bills that they're going to work on ASAP."

Trump suggests changes to House bill

The fast-moving developments came after the House's sweeping coronavirus relief bill, which had been barreling forward in the Senate, had hit a speed bump earlier in the day amid objections by Republicans and a suggestion by President Donald Trump that the Senate would make further changes to the measure.

Also, Trump told reporters that "we may very well be adding something" to the bill, despite his public endorsement of the measure on Friday night.

"I think they may make it even better," Trump said of the Senate. "Look, they're working with the House, working very much in unison like the question before. They're working to only enhance it and make it better and make it fair for everybody. And that's what we're looking to do. So we may go back and forth with the House a little bit, but both will be in a very positive fashion."

The measure passed the House by a 363-40 vote early Saturday morning. Some Republicans objected to several of the provisions in the bill, particularly the paid leave program and its potential impact on businesses throughout the country.

One obstacle was removed Monday evening. The House approved a set of changes to the coronavirus stimulus bill by unanimous consent, clearing the path for the Senate to consider it. Republican Rep. Louie Gohmert of Texas withdrew his objection to the House's 'technical corrections" bill accompanying the coronavirus relief package that passed early Saturday morning. He allowed the measure to advance, saying on the House floor after reviewing the language of the resolution that the changes "make the bill better than it was when it got passed."

On Monday, Gohmert said he raised concerns to both Trump and House Minority Leader Kevin McCarthy, who also supports the House-passed bill.

"The last draft of the 'technical changes' I saw was 87 pages long," Gohmert said in a statement. "I cannot in good conscience give my consent to something that has not been finished or made available to members of Congress before it is up for a vote."

Those objections were apparently settled by Monday evening.

Indeed, a growing number of top Republicans suggested that the Senate should quickly adopt the House plan -- and worry about other proposals for the next response to the crisis.

Sen. John Cornyn, a member of the Senate GOP leadership team, told CNN it's "my inclination" to let the House bill pass unchanged and focus on the next bill.

"It may be the best thing to do would be to make those changes on the next bill because this isn't the last piece of legislation that we are going to be passing related to the coronavirus," the Texas Republican said.

"If we putz around here, we are losing time," said Sen. Shelley Moore Capito, a West Virginia Republican

Sen. Joni Ernst, an Iowa Republican, said she supports moving quickly on the House bill.

The developments come after the Senate scrapped its own recess this week to consider the House bill -- and as senators return to Washington uneasy about the risk they and their staff face by staying in session.

Indeed, prospects appeared to grow slim Monday that the Senate could speed up its schedule this week, but those discussions are expected to during the parties' weekly Tuesday lunches. Democrats are expected to have a conference call for their weekly Tuesday lunch as opposed to meeting in person, an unusual move reflective of how fears over the virus are upending the Senate.

Some Republicans are trying to push for more changes to the bill.

"I don't think the House bill is going to pass the Senate as it is written for one basic problem: It doesn't go far enough and it doesn't go fast enough," GOP Sen. Tom Cotton of Arkansas said on the floor of the Senate, referring to the bill's system for small businesses to get tax credits for paid leave for their displaced employees.

In addition to the coronavirus legislation, the Senate also had to consider a measure to renew key authorities under the Foreign Intelligence Surveillance Act, which expired on Sunday.

The Senate was prepared to take steps to pass the House's bipartisan FISA reauthorization bill, which passed last week, but objections from critics like GOP Sens. Rand Paul of Kentucky and Mike Lee of Utah threatened to eat up several days of the Senate calendar to overcome a filibuster and approve the measure.

Instead, McConnell cut a deal with Lee on Monday that paved the way for the Senate to focus on the coronavirus legislation.

The agreement included a two-and-a-half month extension of the three expired FISA authorities, which passed the Senate by unanimous consent, and agreement to consider amendments from Lee and Paul related to representation for targets of FISA surveillance warrants and limits on searches that can be conducted under the law related to US citizens and the internet.

The short-term extension now has to be cleared in the House by unanimous consent just like the coronavirus changes, meaning any single House member can object and stop it.

But in the Senate, the FISA agreement clears the stage for the chamber to consider the coronavirus legislation — if it can find a way forward amid the disagreements over the bill.

This story has been updated with additional developments Tuesday.