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Negotiators from the House, Senate and White House are close to finalizing an accord on the so-called “phase 3.5” response to the coronavirus pandemic and hope to have an agreement late Monday or Tuesday, Fox News is told.

The response will include $310 billion for the Paycheck Protection Program (PPP), which offers grants to small businesses struggling to make payroll, sources tell Fox News. The $350 billion fund depleted last Thursday.

A separate provision worth $60 billion (including $50 billion in loans and $10 billion in grants) would be allocated to economic disaster loans. Also included: $75 billion for hospitals and $25 billion for coronavirus testing. The total of the package would be north of $470 billion.

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House Democrats were on a caucus-wide conference call late Monday, discussing the next steps. Progressives have some reservations with some provisions in the current bill framework – as does the Congressional Black Caucus. Many Democrats are still concerned by the lack of money for state and local governments in the planned bill. And, some Republicans are signaling they may be reluctant to tackle other measures related to coronavirus, considering the staggering price tag of the measures already in the queue.

Fox News is told that calculus could change if lawmakers believe they now need to respond to the oil market cratering to a historic low.

The stimulus proposal currently under consideration would not include money for state and local governments “nor the Kennedy Center,” said one source. Democrats pushed for the Washington, D.C. performing arts center to receive $25 million in the last stimulus, and Republicans have sought to claw back the money after administrators there conducted layoffs even after obtaining the funds.

However, Republicans and Democrats are still negotiating over state-driven approaches for testing. So far, there are not specific strings as to how hospitals would use the money.

If an accord comes together, the Senate could move the package on Tuesday when it convenes at 4 p.m. ET – with a skeleton crew on hand – likely by unanimous consent, meaning no other senator present in the chamber objects. Then, the bill would move to the House, possibly Wednesday or Thursday.

“We won’t know if we are able to clear it by unanimous consent until we have a product," Senate Majority Leader Mitch McConnell, R-Ky., told Fox News on Monday afternoon. "The talks are continuing. ... And we're getting closer, and hopefully we'll have an agreement of some kind later today.”

Speaking on the Senate floor, McConell criticized Democrats for delaying action on replenishing the PPP. House Speaker Nancy Pelosi, D-Calif., justified the delay on Sunday by saying she wanted more funding for testing and hospitals -- uncontroversial demands that even left-of-center commentators have pointed out that Republicans never opposed.

“It’s now been four days since the Paycheck Protection Program ran out of money. Republicans have been trying to secure more funding for this critical program for a week and a half now,” McConnell said. “At this hour, our Democratic colleagues are still prolonging their discussions with the administration, so the Senate, regretfully, will not be able to pass more funding for Americans’ paychecks today."

McConnell added: “However, since this is so urgent, I have asked that the Senate meet again tomorrow in a new session that was not previously scheduled, and the Democratic Leader has agreed to my request. Colleagues it’s past time, past time to get this done for the country.”

President Trump told reporters at Sunday's White House coronavirus press briefing that a PPP deal was imminent. "We're looking to help hospitals, rural hospitals as part of the package," Trump said. "We could have an answer tomorrow." At Monday's briefing, Trump said he hoped for a Senate vote on Tuesday.

The office of House Majority Leader Steny Hoyer, D-Md., signaled over the weekend that House members should be ready to come back Wednesday. Hoyer’s office also indicated they could take a “roll call vote.” On Monday afternoon, Hoyer told Democrats that they are hoping to vote Wednesday, but that it could slip to Thursday.

Fox News is told that the “appetite for a roll call vote is higher” than the vote taken in late March on the phase 3 coronavirus bill, totaling $2.2 trillion.

It’s possible the House Rules Committee may have to meet Wednesday or later in the week to prep the House to handle the measure sent over from the Senate. The Rules Committee is prepared to meet in a larger space if necessary. But, it is generally thought the Rules Committee would not have to meet on the "phase 3.5" bill. The House could advance this plan as a “suspension” bill. That means the House expedites the process, but needs a two-thirds supermajority to pass.

If there is a roll call vote in the House, Fox News is told the House would likely clump the entire House membership into six or seven clumps alphabetically and have members file into the chamber in groups. This could consume several hours – but help members exercise social distancing. One source told Fox News, however, that the House would be lucky to get 280 members there.

Fox News is told a big question is if the House could vote on other items. One issue could be a vote to launch the select committee to review the pandemic response engineered by Pelosi. Most Republicans will oppose the creation of the select committee. That could entail action from the Rules Committee and a possible vote.

There is also discussion about the concept of “remote” or “proxy” voting. It’s unclear if there is any energy to implement either of those strategies. Pelosi has said a vote to change the rules temporarily for voting must be “bipartisan.”

That could mean Pelosi doesn’t have to move on the proxy or remote voting issue so long as there no consensus on the next bill. It could also mean that she is unwilling to commit because she doesn’t have GOP buy-in. It could also mean that the Speaker doesn’t have consensus on her side for proxy or remote voting.

That said, Democrats have been hammered by Republicans for blocking the next round of funding for small businesses. The theory is that the sides have to get a deal this week because the cupboard for the small business program went bare last week. So, as one senior House Democratic aid said to Fox, such a deadline “tends to focus the mind.”

Regardless, a plan for remote or proxy voting would likely have to go to the Rules Committee before the House floor.

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Meanwhile, oil futures plunged below zero for the first time on Monday as demand for energy collapses amid the coronavirus pandemic and traders sought to avoid owning crude with nowhere to store it. Fox News is told the development could impact the contents of the upcoming stimulus package.

Stocks were also slipping on Wall Street in afternoon trading, with the S&P 500 down 1.2 percent, but the market’s most dramatic action by far was in oil, where benchmark U.S. crude for May delivery plummeted to negative $35.20, as of 2:30 p.m. ET. It was nearly $60 at the start of the year, before business-shutdown orders swept the world and idled factories, offices and automobiles.

Much of the drop was chalked up to technical reasons — the May delivery contract is close to expiring so its trading volume was light, which can exacerbate swings. But prices for deliveries even further into the future, which were seeing larger trading volumes, also plunged. Demand for oil has collapsed so much that facilities for storing crude are nearly full.

“The government can declare whatever they want in terms of encouraging people to get out and do stuff,” said Willie Delwiche, investment strategist at Baird. “Whether or not broad swaths of society do that remains to be seen. It’s going to take seeing people start to get out and do stuff again. That will be the necessary positive development, not just declaring getting things open.”

Fox News' Jason Donner and Marisa Shultz, as well as The Associated Press, contributed to this report.