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WASHINGTON — The second day of a government shutdown began with Congress no closer to an agreement as the stand-off continued without a clear resolution in sight.

The Senate's leaders, Republican Mitch McConnell and Democrat Chuck Schumer, did not speak at all on Saturday, according to a spokesman for Schumer, indicating how dug in each side is. The Senate is slated to return to session at 1 p.m. ET Sunday, with the House returning an hour later.

As the stalemate persisted Saturday, groups of senators met on their own initiative to search for a solution. Sens. Lindsey Graham, R-S.C., and Jeff Flake, R-Ariz., shuttled between the offices of McConnell and Schumer late Saturday evening, acting as liaisons.

McConnell scheduled a vote for 1 a.m. Monday on the only proposal on the table as of now: a three-week spending bill. It's unclear if the measure obtained the 60 votes necessary in the Senate.

Speaker Paul Ryan said on CBS's "Face the Nation" Sunday that the House is prepared to accept the continuing resolution if the Senate is able to pass it.

"We passed a bill keeping things funded to Feb. 16," Ryan said. "[McConnell] is going to bring up a bill to keep things funded until Feb. 8. We have agreed that we would accept that in the House, and so we will see sometime today whether or not they have the votes for that and that's really where we are right now."

But the real stalemate remains in the Senate.

Related: Tom Cotton says his support for immigration deal won't hinge on Trump

The politics of which party is blamed for a government shutdown are in full effect as President Donald Trump's campaign team released a politically explosive and divisive ad and Trump referred to the 2018 midterm elections in a series of tweets.

A breakdown of trust between negotiators, particularly Schumer and the president, has played a major role in the current stalemate.

"Negotiating with President Trump is like negotiating with Jell-O," Schumer said Saturday, a day after negotiations broke down after a meeting with Trump.

The latest from Capitol Hill

What’s happening : Senators appear no closer to an agreement on how to open the government.

: Senators appear no closer to an agreement on how to open the government. What’s on the table : A short-term spending bill that lasts just three weeks, until Feb. 8. It would extend CHIP for six years and suspend some Obamacare taxes.

: A short-term spending bill that lasts just three weeks, until Feb. 8. It would extend CHIP for six years and suspend some Obamacare taxes. Why Feb. 8? Sen. Lindsey Graham argues that three weeks is short enough to keep momentum on negotiations over a series of issues: DACA, CHIP, disaster aid, spending levels for a two-year spending bill and Alexander-Murray insurance market stabilization.

Sen. Lindsey Graham argues that three weeks is short enough to keep momentum on negotiations over a series of issues: DACA, CHIP, disaster aid, spending levels for a two-year spending bill and Alexander-Murray insurance market stabilization. What's next? Graham says that McConnell promised him a vote on a bipartisan immigration bill, even if the president doesn’t support it, if the talks between Sens. McCarthy, Cornyn, Durbin and Hoyer break down.

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The senator said that he offered Trump his border wall as a concession to seek movement on "Dreamers," people who were brought to the U.S. illegally as children, but later in the day on Friday Trump backed off the discussions.

Office of Management and Budget Director Mick Mulvaney said at a White House press briefing on Saturday that Schumer's offer for the border wall was $1.6 billion, an amount that Trump had previously rejected as not enough.

Schumer fired back, telling NBC News that Mulvaney wasn't in the room and "doesn't know the truth."

"[Trump] put a number on the table and we took it," Schumer told NBC News. Schumer's spokesman, Matt House, said the number was "way, way north" of $1.6 billion.

McConnell said Schumer is blaming everyone but himself.

"Like the president, like the House, and like a bipartisan majority of senators, the American people want long-term solutions on immigration policy. On government spending. On all the major issues we've been discussing for months — and will continue to discuss," McConnell said.

While Trump was on the phone with aides throughout Saturday, some Democrats are starting to quietly wonder where this goes.

"I’m concerned that we don't have an exit strategy," one Democratic aide for a liberal senator told NBC News. "I think that it seems naive to think that Republicans will do the right thing here and compromise."

"This is a question of who's going to flinch first," the aide said.

Meanwhile, as leadership struggles to an agreement, a bipartisan group of lawmakers led by Sen. Susan Collins, R-Maine, organized a meeting Saturday afternoon to discuss a solution. Sen. Joe Manchin, D-W.V., said afterward that the group of about 15 senators are hoping to present three plans to leadership.

Many Democrats say a three or four week spending bill, as floated by McConnell early Saturday, is just a stalling tactic to avoid negotiating on a myriad of issues, especially DACA, but also disaster aid and government spending levels, both parties have yet to agree upon.

But Graham, who was one of four Republicans to vote against the funding bill Friday, is attempting to persuade his colleagues to consider the short-term resolution.

"After my discussions with numerous senators on both sides of the last night it is clear to me a commitment to move to immigration after February 8th is the key to ending the government shutdown and finding resolution on all the outstanding issues," Graham said on Twitter, part of a string of tweets outlining his support for the offer.

Leader McConnell mentioned last night he was willing to make a proposal for a CR through February 8 and seek resolution on immigration, disaster relief, military and government funding, CHIP, and other health care related issues. I would support such a proposal. — Lindsey Graham (@LindseyGrahamSC) January 20, 2018

But many Democrats say they want more than McConnell’s word that he’s “willing” to move forward on such issues. They say they want tangible progress.

“February is longer than what I want," said Sen. Tim Kaine, D-Va., who represents hundreds of thousands of federal workers who live in Virginia. "Depending on the commitments they make, that could be good. Well, if it’s just Feb. 8 and just another delay, that’s not going to be acceptable.”

Trump used his early-morning tweets Saturday to claim that Democrats are harming the military in favor of protecting Dreamers.

Democrats are holding our Military hostage over their desire to have unchecked illegal immigration. Can’t let that happen! — Donald J. Trump (@realDonaldTrump) January 20, 2018

And he made it a campaign issue as well, tweeting that the 60-vote threshold in the Senate to pass legislation "is why we need to win more Republicans in '18." His campaign also released a campaign ad highlighting an undocumented immigrant charged with killing two police officers.

Sen. Tammy Duckworth, D-Ill., who is an Iraq war veteran, on Saturday slammed the president’s tweet that accused Democrats of holding the military “hostage.”

"I will not be lectured about what our military needs by a five-deferment draft dodger," Duckworth said on the Senate floor. She said that if Trump truly cared about the military "he would stop hiding behind his Twitter account, stop blaming everyone else."

"And he can tell his party — a party that controls the House, the White House, and the Senate — to do their job," she said.

Trump, for his part, has postponed a planned trip to his private resort in Palm Beach, Florida, where he had been set to attend a pricey anniversary party for his presidency.

According to a source familiar with the planning of the event, the fundraiser at Mar-a-Lago will still go on as planned and the president's elder sons — Eric Trump and Donald Trump Jr. — will attend.

But the White House says the president will not travel to Florida until some kind of deal is reached to reopen the government.