Government shutdown: Congress seeks a deal in flurry of negotiations -- live updates

House and Senate lawmakers engaged in another round of negotiations to end the government shutdown as the impasse continued into the second day, with both chambers reconvening with the goal of striking a deal on spending and immigration that would reopen federal agencies ahead of the work week.

The prospects of an agreement seemed slightly less daunting Sunday morning as the fragile outlines of a potential deal began to take shape.

On CBS News' "Face the Nation," House Speaker Paul Ryan said the lower chamber has agreed to accept a short-term deal that would fund the government through Feb. 8 if the Senate is able to pass such a bill. Senate Majority Leader Mitch McConnell has said the Senate will vote on a Feb. 8 bill at 1 a.m. Monday.

Senate Democrats are unlikely to support a revised bill without concessions on immigration from the GOP. On Saturday, Republican Sen. Lindsey Graham of South Carolina hinted at one possibility: a bill that would fund the government in exchange for a commitment to move onto immigration after Feb. 8.

Where things stand on Day 2

• Ryan says House would support short-term deal
• House and Senate to reconvene in rare Sunday session
• Trump calls on Senate to abolish filibuster
• McConnell schedules Senate vote for 1 a.m. on continuing resolution

"After extensive discussions with Senators, on both sides of the aisle, I believe such a proposal would pass if it was understood that after February 8, the Senate would move to an immigration debate with an open amendment process if no agreement has been reached with the White House and House of Representatives," Graham said in a statement Saturday afternoon. 

Graham predicted that a "breakthrough" would happen Sunday night as moderate lawmakers met with GOP leaders to discuss the impasse.

On Saturday, Republicans and Democrats failed to reach an agreement as federal agencies began implementing shutdown procedures. McConnell told reporters that lawmakers would be "right back at it" for "as long as it takes."

"We will keep at this until Democrats end their extraordinary filibuster of government funding and children's healthcare, and allow a bipartisan majority of Senators to reopen the federal government for all Americans and get Congress back on track," McConnell said Saturday on the Senate floor. Senate Democrats say they will not support a funding resolution that does not include protections for immigrants brought to the U.S. as children under the DACA program and spending for disaster relief.A

Follow along below for updates on the shutdown. All times Eastern unless otherwise noted.

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3:17 p.m.: Rep. Kevin McCarthy: Shutdown gets "more serious" on Monday

House Majority Leader Kevin McCarthy told reporters that he's hoping the Senate will move something forward today because the shutdown becomes more serious Monday. "We're just hoping that calmer heads prevail, that in the Senate they see the action and move forward," he said.

He added, "If they take a week off the continuing resolution, we'll take that. And keep the government working, make sure Monday morning everything is up and running."

With regards to immigration talks, McCarthy said that negotiations could continue once the government reopens.

"We were having great meetings prior but unfortunately, the government got shut down. As soon as it opens, we'll go right back to our meetings," he said.

3:00 p.m.: Lawmakers talk shutdown negotiations 

Sen. Bill Nelson, D-Florida, told reporters on Capitol Hill that he's "optimistic" about current negotiations over the funding of the federal government, but conceded that there's a "long way to go before we get there." 

Meanwhile, Sen. Bob Corker, R-Tennessee, said that lawmakers were "so close, it's ridiculous."

Sen. Lamar Alexander, R-Tennessee, said a shutdown "should never ever be used as a bargaining chip for any issue period," and denounced it as "chemical warfare" that should be "banned."

"It hurts taxpayers, it hurts the military, it hurts our country."

Alexander said that there were a number of lawmakers who are "ready to go to work on funding for the military over two years, funding for the National Institutes of Health. We're close to agreement on that."

"The Alexander-Murray proposal to reduce insurance rates to children's health insurance plan, DACA, disaster aid. That's all there. We're on the 10 yard line," he added.

1:56 p.m.: Graham says Stephen Miller to blame for stalemate

Graham told reporters on Capitol Hill that he thinks Mr. Trump has his "heart right" on the issue of immigration reform, but says "every time we have a proposal it is only yanked back by staff members." 

"As long as Stephen Miller is in charge of negotiating immigration we are going nowhere. He's been an outlier for years," added Graham, referring to the White House senior policy adviser, an immigration hardliner. Graham predicted that "there will be a breakthrough tonight, if there's going to be a one it will be tonight."

1:35 p.m.: Democrats hold press conference on status of shutdown

House Minority Leader Nancy Pelosi says that Democrats could sign onto an agreement "in an hour," calling on the president to come to the negotiating table to resolve the shutdown. Democrats said they were committed to do what they can to ensure military members receive their paychecks during the government shutdown. 

Government shutdown: Congress seeks a deal in flurry of negotiations -- live updates