Washington (CNN) The Senate is scheduled to vote Monday on whether to re-open the government following Majority Leader Mitch McConnell's pledge to take up legislation to address an expiring immigration program in hopes of enticing enough Democrats to come on board.

The vote was moved from 1 a.m. ET Monday to noon after it became clear Democrats would block the spending bill over disagreements on a variety of issues, most notably what do about young people impacted by the Deferred Action for Childhood Arrivals program

Senate Majority Whip John Cornyn of Texas said he thought Senate Minority Leader Chuck Schumer agreed to push back the vote to give his caucus "a chance to chew" on a GOP proposal to break the impasse.

"It's better to have a successful vote tomorrow at noon than a failed vote tonight," Cornyn told reporters.

It's not clear if McConnell's pledge will be enough to win the 60 votes need to pass a continuing resolution, but at least one previous no vote -- Sen. Jeff Flake, an Arizona Republican -- said he was now a yes. Republicans only control 51 seats in the chamber and therefore need Democrats to vote to pass it.

A top Democratic leadership aide disputed Cornyn's assertion and said unless Republicans make significant changes to their offer, Democrats will likely reject it when the vote comes at noon. The Democratic aide did say, however, that progress was made in the lengthy negotiations that took place late Sunday night. But more talks are needed.

Two Democratic sources said late Sunday night that, as of that time, they did not expect the needed 60 votes to pass the continuing resolution would come at noon Monday.

If there is progress overnight or if the mood shifts after meetings Monday morning, the situation could change, the Democratic sources said.

McConnell said from the Senate floor Sunday night that "should these issues not be resolved by the time the funding bill before us expires on February 8, 2018, assuming that the government remains open, it would be my intention to proceed to legislation that would address (the expiring Deferred Action for Childhood Arrivals program), border security, and related issues."

A meeting earlier Sunday between McConnell and Minority Leader Chuck Schumer did not lead to a spending deal involving when to hold votes on the DACA program and whether to support the three-week funding solution, two sources told CNN. A Senate Democratic source said Sunday that Schumer rejected McConnell's offer because it wasn't a firm commitment to get those measures on the President's desk.

At a bipartisan meeting with senators at the office of Republican Sen. Susan Collins of Maine on Sunday, a sticking point was over the date of when to hold the DACA vote: whether it would occur by the February 8 deadline of when the proposed continuing resolution would expire.

The Republicans in the bipartisan group want to wait until after February 8, but Democratic senators want to tie DACA to the next spending bill.

Despite Schumer's rejection of the deal, one senior GOP aide involved in the talks said Republican leaders think they have a shot of picking off enough Democrats to move forward. "We're close," the aide said.

With Flake and and South Carolina Republican Lindsey Graham, who voted against the short-term funding plan on Friday, now as "yes" votes, Republicans will need seven more Democrats to get to 60. Graham has consistently told leaders behind closed doors he thinks he has peeled off a handful of Democrats.

While McConnell's commitment falls far short of what the vast majority of Democrats want, there's some GOP hope they can get enough to move forward -- and notably split the caucus.

"This is their off ramp," the aide said. "We'll see if they dig in or if they want a way out."

Blame game continues

McConnell and Schumer each took to the chamber floor on Sunday to trade blame for the shutdown, both accusing the other side of taking hostages.

But in contrast to the sharp speeches from the respective party leaders, a few Republican senators sounded somewhat confident that they would receive enough support from Democrats to reopen the government on a short-term basis without a simultaneous deal on immigration.

Graham and Cornyn both predicted they could get enough support from Democrats in the vote early Monday to pass a three-week extension.

Graham said that the White House was not leading on the situation and that White House staffers were serving the President poorly.

Graham specifically slammed White House senior policy adviser Stephen Miller for his influence on the President.

"Every time we have a proposal, it's only yanked back by staff members," he told reporters. "And as long as Stephen Miller is in charge of negotiating immigration, we're going nowhere. He's been an outlier for years."

White House deputy press secretary Hogan Gidley responded to Graham's comments, calling him the "outlier" in the Senate.

"As long as Sen. Graham chooses to support legislation that sides with people in this country illegally and unlawfully instead of our own American citizens, we're going nowhere," he said. "He's been an outlier for years."

Also Sunday afternoon, Democratic Sen. Tammy Duckworth of Illinois took to the Senate floor to call for a measure in the meantime to assure pay for military personnel and for military families to receive death benefits despite the shutdown.

"Let's at least take the simple, commonsense step that we all agree on," Duckworth said. "Let us remove any possibility military pay and even worse, military death benefits will be used and held hostage as political leverage."

Dueling speeches

During his speech early Sunday afternoon, McConnell said Democratic demands on permanent protections for DACA recipients were not an emergency given that the program is not set to expire until March. He said the expired Children's Health Insurance Program and other government programs were of more immediate concern.

"All of these other things are an emergency," McConnell said. "The one non-emergency issue that our friends on the other side are trying to shoehorn into this discussion doesn't reach that status of emergency until March."

McConnell said he supported the right of the Democratic minority to filibuster "from an institutional point of view," but maintained Schumer's use of it was unproductive and that the Democrats should relent.

Schumer, meanwhile, pinned blame for the shutdown on Republicans in general and the President foremost.

"Congressional leaders tell me to negotiate with President Trump," Schumer said. "President Trump tells me to figure it out with the congressional leaders."

Schumer said he offered Trump a compromise on immigration that included funding his proposed border wall with Mexico, but that ultimately Trump and the Republicans had not moved on his proposal.

"Because the President campaigned on the wall, even though he said it would be paid by Mexico, and demands the wall, for the sake of compromise, for the sake of coming together, I offered it," Schumer said. "Despite what some people are saying on TV -- and mind you these are folks not in the room during discussion -- that is exactly what happened. The President picked a number for a wall. I accepted it."

The White House in turn denied Schumer's claims about the Friday meeting he had with Trump.

"Sen. Schumer's memory is hazy because his account of Friday's meeting is false," White House press secretary Sarah Sanders said in a statement. "And the President's position is clear: We will not negotiate on the status of unlawful immigrants while Sen. Schumer and the Democrats hold the government for millions of Americans and our troops hostage."

She also said Sunday afternoon that Trump had spoken with Veterans Affairs Secretary David Shulkin and Homeland Security Secretary Kirstjen Nielsen about the impact of government shutdown, and that Trump and White House chief of staff had each spoken with members of Republican congressional leadership.

White House budget director Mick Mulvaney said in an interview on CNN's "State of the Union" Sunday morning that Schumer had not exactly offered the White House what they wanted on the wall, arguing Democrats had offered "to authorize" the wall, but not to appropriate funds for it.

Nuclear option

Earlier Sunday, the White House called for Senate Republicans to change the chamber's rules to resolve the funding impasse as the government shutdown continued into its second day.

Trump tweeted his call for Senate Majority Leader Mitch McConnell to invoke the so-called nuclear option and thereby remove leverage for Senate Democrats.

"Great to see how hard Republicans are fighting for our Military and Safety at the Border. The Dems just want illegal immigrants to pour into our nation unchecked. If stalemate continues, Republicans should go to 51% (Nuclear Option) and vote on real, long term budget, no C.R.'s!" Trump tweeted.

Great to see how hard Republicans are fighting for our Military and Safety at the Border. The Dems just want illegal immigrants to pour into our nation unchecked. If stalemate continues, Republicans should go to 51% (Nuclear Option) and vote on real, long term budget, no C.R.'s! — Donald J. Trump (@realDonaldTrump) January 21, 2018

A spokesman for McConnell said in response to the tweet that the Senate Republican Conference does not support changing the 60-vote rule, a reiteration of Republican Senate leadership's already-stated opposition to the move Trump has called for over the past year.

Senate rules impose a threshold of 60 votes to break a filibuster, and Senate Republicans currently hold a slim majority of 51 votes, meaning even if they can unite their members, they need nine more votes to end debate. The White House is calling for the Senate to change its rules and move the threshold to a simple majority of 51 votes.

Eliminating the 60-vote threshold to break a legislative filibuster would remove significant powers for the minority party in the Senate, and party leaders have been reluctant to do so in the past because of the consequences it would pose when their party returns to the minority.

During his CNN interview, Mulvaney said eliminating the filibuster would be one avenue they back to ending the shutdown.

"There's a bunch of different ways to fix this," Mulvaney said. "We just want it to get fixed."