Senate Democrats and the Biden administration are forging ahead with President Joe Biden's $1.9 trillion COVID-19 relief bill, which the House passed over the weekend, Senate Majority Leader Chuck Schumer said on the Senate floor.

Now comes the matter of hammering out the details.

President Biden, speaking to Senate Democrats on Tuesday, urged them to pass the bill swiftly, highlighting widespread bipartisan support for the bill, "including from a bipartisan majority of the American people, state and local leaders, labor, and the business community," and urged the need to get the bill passed to reopen schools and speed up vaccinations, according to a readout from the White House.

"With the Senate preparing to take up the Rescue Plan, the President urged the caucus to remain united in its commitment to passing a package big enough to meet the economic and health crises we face, and to act swiftly to pass it before unemployment benefits for around 11 million Americans start expiring on March 14," the White House said in a statement recapping the call.

"The fact is that $600 is not enough," Biden wrote on Twitter on Tuesday. "The Senate needs to pass the American Rescue Plan and finish the job of delivering $2,000 in direct relief."

"The Senate will take up the American Rescue Plan this week," Schumer said Monday. "I expect a hardy debate and some late nights, but the American people sent us here with a job to do ... to end through action the greatest health crisis our country has faced in a century."

Senate Minority Leader Mitch McConnell (R-KY) slammed the bill, calling it a "bonanza of partisan spending they're calling a pandemic rescue package" and calling an extension of federal unemployment benefits included in the bill "a premium to stay home that would extend well into a recovery where job growth and rehiring would be pivotal."

On Tuesday, McConnell doubled down on his criticism: "We'll be fighting in every way that we can," he said. "It is my hope that at the end Senate Republicans will unanimously oppose it."

The bill provides $1,400 payments to individuals plus hundreds of billions of dollars for schools and colleges, COVID-19 vaccines and testing, mass transit systems, renters and small businesses, as well as money for child care, and an expansion to tax breaks for families with children.

Senate Democrats may reshape the $350 billion the bill provides for state and local governments. They also might extend its fresh round of emergency unemployment benefits, which would be $400 weekly, through September instead of August, as the House approved.

Democrats are hashing out the finer details of the bill.

Some Democratic moderates including Sen. Joe Manchin (D-WV) want to trim the unemployment figure back to $300, arguing more weekly money would discourage people from returning to work.

“It would be awful for the doors to open up and there’s no one working,” Manchin said. But party leaders and progressives oppose trimming those benefits, and Manchin conceded he didn’t know that his side would prevail.

Despite the huge leverage every Democrat has because all their votes are needed, none has so far threatened to sink the legislation if they don’t get their way.

Manchin and Sen. Jon Tester (D-MT), said they want spending to be better “targeted,” which Manchin said meant “helping the people that need help the most.” Republicans have said the legislation is too costly and spends money needlessly.

Sen. Angus King (I-ME) said he wants the bill’s $350 billion for state and local governments to specify minimum amounts for municipal governments. He also said he wants perhaps $50 billion to improve broadband coverage.

What the bill won't include is a $15 increase to the minimum wage.

The move comes as Progressive Caucus Deputy Whip Rep. Ro Khanna (D-CA) and 22 Democratic House lawmakers sent a letter to President Biden and Vice President Harris, urging them to override the Senate parliamentarian's ruling regarding the federal minimum wage increase.

Elizabeth MacDonough, the U.S. Senate's nonpartisan parliamentarian, has ruled that an increase of the federal minimum wage to $15 can not be included in the COVID-19 relief bill, dealing a fatal blow to the prospect of increasing the federal minimum wage, which has been set at $7.25 since 2009.

Sens. Bernie Sanders (I-VT) and Ron Wyden (D-OR) said that they were considering a tax on large companies that did not increase their minimum wage, but Democratic Senate aides speaking on the condition of anonymity said that they are dropping efforts to include a $15 minimum wage increase in the bill at all.  The Washington Post was the first to report the decision.

A number of progressives and outside groups have urged Biden and Harris to ignore the advice of the parliamentarian, which would require the support of 50 Senators. It would take 60 votes to overrule Harris' decision, but such a move has never been done under the reconciliation process.

"This ruling is a bridge too far. We’ve been asked, politely but firmly, to compromise on nearly all of our principles & goals. Not this time," the letter said. "If we don’t overrule the Senate parliamentarian, we are condoning poverty wages for millions of Americans. That’s why I’m leading my colleagues in urging the Biden Administration to lean on the clear precedent and overrule this misguided decision."

White House Press Secretary Jen Psaki told reporters that the Biden administration would not pursue such a course of action: "That’s not an action we intend to take," Psaki said, adding that they do not have the support of 50 Senators for such a move.

"The decision for the vice president to vote to overrule or to take a step to overrule is not a simple decision. It would also require 50 votes," she said, "And the president and the vice president both respect the history of the Senate."

Illinois Sen. Dick Durbin, the no. 2 Democrat in the Senate, also poured water on the idea. "I don’t think that’s going to work," he said, giving hope to future endeavors to increase the minimum wage. "I hope that we think very seriously about dealing with the minimum wage in a different venue."

Progressives have aggressively pushed for an increase to the minimum wage since taking the Senate and White House, but moderate Democrats – namely Sens. Manchin (D-WV) and Kyrsten Sinema (D-AZ) – had previously objected to including the minimum wage hike in the $1.9 trillion COVID-19 relief bill.

Republicans have been staunch opponents of the increased minimum wage, believing that it would cost jobs and impact businesses; they have also opposed the relief bill, saying it is too expensive.

Removal of the minimum wage hike might ultimately make the bill easier to pass, as it would likely all but guarantee Manchin and Sinema's support, giving Democrats the 50 votes they need to pass President Biden's top legislative priority of his early presidency.

Sen. Sanders, the Budget Chairman, has said he will force a vote on an amendment restoring the increase to $15, though it seems virtually certain to lose.

House Speaker Nancy Pelosi (D-CA) said that the House will "absolutely" pass the COVID-19 relief bill, with or without minimum wage.

Psaki said that Biden still supports increasing the minimum wage, but did not commit to a specific plan about how to carry it out: "We don’t have a clear answer on what that looks like at this point. It just remains a commitment and something he will use his political capital to get done."

"The president’s focus this week and in coming weeks, until it’s passed, is on the American Rescue Plan," Psaki added. "It’s absolutely critical Congress act, and we certainly hope they do that as quickly as possible."

The Associated Press contributed to this report.