WASHINGTON — The White House and congressional Democrats on Sunday closed in on an agreement for a $450 billion economic relief package to replenish a depleted emergency fund for small businesses and to expand coronavirus testing around the country, with votes on the measure possible early this week.

Treasury Secretary Steven Mnuchin described the broad outlines of the package in an appearance on CNN on Sunday. The agreement would include $300 billion to replenish the emergency fund, called the Paycheck Protection Program; $50 billion for the Small Business Administration’s disaster relief fund; $75 billion for hospitals and $25 billion for testing.

Speaker Nancy Pelosi and Sen. Chuck Schumer, D-N.Y., the minority leader, said in separate television appearances Sunday morning that a deal appeared to be in the offing.

“We’ve made very good progress, and I’m very hopeful we could come to an agreement tonight or early tomorrow morning,” Schumer said, appearing shortly after Mnuchin on CNN’s “State of the Union.” He said the White House was “going along with” some of the Democrats’ requests, “so we feel pretty good.”

Mnuchin said President Donald Trump approved of the framework, and the president himself expressed optimism Sunday about an agreement. “We are very close to a deal,” Trump said.

Mnuchin said he hoped that the Senate could vote on the bill as early as Monday and the House on Tuesday.

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That would represent a significant breakthrough after nearly two weeks of stalemate over the bill, even as the $349 billion small-business fund ran dry Thursday with many applicants still in line, a move that risked adding more bankruptcies, business failures and job losses to an already stunning economic toll.

That timeline, however, could be optimistic, and the path ahead is complicated. With lawmakers scattered across the country, many in states that are restricting travel, House and Senate leaders will most likely try to approve any agreement during procedural sessions this week as opposed to bringing their rank and file back to the Capitol to vote. But during procedural sessions, any one lawmaker could object, delaying final passage.

The money for hospitals and testing in the package that Mnuchin outlined was a significant concession to Democrats, who were standing in the way of a quick and stand-alone infusion of cash to the Paycheck Protection Program, which offers forgivable loans to small businesses to create incentives for them to keep employees on their payroll.

Democrats had also wanted to couple an infusion for the small-business program with more money for states and cities. But Mnuchin said such funds would be included in a future relief package.

Later Sunday, Sens. Bill Cassidy, R-La., and Bob Menendez, D-N.J., announced what they described as “a major bipartisan breakthrough” to deliver funds to states and communities on the front lines of the fight against the coronavirus.

Their proposal would make counties and towns with 50,000 or more people eligible for federal dollars; the current population threshold is 500,000. The senators said they would introduce the plan when the Senate convened.

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“Sen. Menendez’s state and mine were hit hard by the COVID-19 epidemic,” Cassidy, a gastroenterologist, said in a statement, adding that the two “worked hard to make sure state and local governments can maintain essential services necessary for employees and employers to survive.”

That bipartisan effort stood in stark contrast to the partisan warfare that has enveloped the talks over the small-business aid since the start. On Sunday, Trump attacked Pelosi on Twitter on Sunday as “an inherently ‘dumb’ person” and predicted that she would be “overthrown” as speaker, “either by inside or out.”

That capped a week that Republicans and Democrats on Capitol Hill spent trading barbs. Republicans, who argued that there was no need to add money for hospitals and testing when it had not yet run out, accused Democrats of holding small businesses hostage while unemployment numbers soared.

“I cannot understand, after watching another 5 million get unemployed, how Speaker Pelosi continues to say no,” Rep. Kevin McCarthy, R-Calif., the minority leader, said Thursday on a conference call with reporters.

Republicans have also expressed strong opposition to adding money for states and municipalities, saying that Democrats have pushed for unrestricted funds, not related to the coronavirus, that would effectively subsidize bad fiscal decisions that occurred before the pandemic. That has been a red line for Republicans throughout the talks.

But after the funding for the Paycheck Protection Program lapsed, Republicans expressed the first hints of openness to accepting at least some of the Democrats’ demands. In an interview with Politico on Friday, McCarthy said he was “fine with doing some hospital” funding as part of a package to shore up the program.

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Some Republicans, though, have expressed skepticism about Mnuchin, whom they see as accommodating to Democrats. Asked Thursday about how a deal that included hospital money would be received, Sen. Mitch McConnell, R-Ky., the majority leader, remained noncommittal, saying only, “We’d take a look at it.”

Mnuchin also said Sunday that he was hopeful that the economy could rebound in a matter of months rather than years. He said that he hoped the extraordinary efforts the government had taken to encourage businesses to keep workers on their payrolls would prevent the jobless rate from reaching 20%.

McConnell hosted a call Sunday with Mnuchin, Trump, Republican senators and Mark Meadows, the White House chief of staff, about the ongoing negotiations. McConnell said that additional funds for state and local governments as well as more money for food assistance would not be included in the final package, according to an aide for a Republican leader who requested anonymity to disclose details of a private phone call.

Mnuchin added that some of the unresolved items were related to funding for testing and that he would be conferring with McConnell; Sen. Lamar Alexander of Tennessee, chairman of the Senate Health Committee; and Sen. Roy Blunt of Missouri, chairman of the Senate Rules Committee, to resolve the issues to ensure swift passage.

Democrats are calling the current package Stimulus 3.5, in reference to the three bills that came before it. In the House, Pelosi and her committee leaders are already working on elements of a fourth package, although it is unclear what that will look like.

Once negotiations on the current bill draw to a close, Pelosi and McConnell will have to figure out a schedule for voting. One possibility is that the measure could be approved by voice vote, which would spare members the necessity of returning to the Capitol.

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But a single lawmaker could stand in the way of such a maneuver, which is what happened last month when the House took up the $2 trillion stimulus package that created the paycheck program. Rep. Thomas Massie, R-Ky., thwarted the effort by demanding a quorum and insisting his colleagues show up in person.

Anticipating a repeat of that episode, Pelosi said last week that she would back a system of remote voting by proxy — a major shift for the speaker and one that would break with centuries of tradition of voting in person. To do so would require a change in House rules, which itself would require an in-person vote.

Rep. Jim McGovern, D-Mass., chairman of the Rules Committee, who has been deputized by Pelosi to examine alternative methods of voting, proposed the proxy voting plan last week. It would allow lawmakers who could not travel to the Capitol because of the pandemic to give explicit instructions on each vote to a colleague who would be authorized to act on their behalf.

In an appearance on “Fox News Sunday,” Pelosi said she wanted to know what Republicans thought before moving ahead with the plan.

“We want to keep the faith on both sides,” she said.

In an interview Friday, Massie said — somewhat surprisingly — that he would not object to forgoing a floor vote or changing rules to permit proxy voting so long as the process was sufficiently “transparent.” Massie said the rule should require that the names of those who voted in person and who voted by proxy be made public along with the names of those whom absent lawmakers had authorized to vote on their behalf.