Day two of the third US government shutdown in a year saw hundreds of thousands of federal employees without pay, national parks closed and Donald Trump stubbornly refusing to accept the blame for an event he previously declared he would be “proud” to cause.

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Acting White House chief of staff Mick Mulvaney said the administration was waiting to hear from congressional leaders about an offer of a deal. But he also said the shutdown, caused by a standoff over border security funding, could “very likely” extend into 2019.

A senior Republican senator called the president’s behaviour in provoking and prolonging the shutdown “useless” and “puerile”.

The Senate adjourned on Saturday, majority leader Mitch McConnell citing a Republican House bill that included more than $5bn for Trump’s border wall as he passed the ball to Democrats and the White House. That meant the shutdown would continue until at least Thursday, after the Christmas holiday. Congress gathers again on 3 January.

Mulvaney told Fox News Sunday it was “very possible” the shutdown could reach the new year.

Speaking to ABC’s This Week, Mulvaney said he and Mike Pence went to Capitol Hill “late yesterday afternoon and we’re waiting to hear back”. He also said Trump had “made it very clear” that he was “willing to discuss a larger immigration solution”.

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Bob Corker, the Tennessee Republican who chairs the Senate foreign relations committee and will retire at the end of the year, did not express such optimism. He said Trump had contrived the shutdown as a campaign issue.

“This is a made-up fight so the president can look like he’s fighting,” Corker told CNN’s State of the Union, adding that precedent showed Democrats would back much larger spending for border security in return for reform, such as to the status of Dreamers, young undocumented migrants brought to the US as children – just not a wall.

“This is something that is useless, it’s spectacle, it’s puerile,” Corker said.

Mulvaney confirmed Corker’s point about why Trump prompted the face-off. The president was “proud to have this fight”, he told Fox, adding that “chaos” was “what Washington looks like when you have a president who refuses to sort of go along to get along”.

At the White House on Saturday, Trump lunched with rightwingers including House Freedom Caucus chiefs Mark Meadows of North Carolina and Jim Jordan of Ohio. No Republican leaders or Democrats, needed for any deal, were present.

Trump will remain in Washington for Christmas, his Florida vacation cancelled, first lady Melania Trump forced to fly back into town. On Sunday he continued to use Twitter to claim the Democrats were to blame for a shutdown he said last week he wanted.

The House bill that would have given him his required funds was a purely political gambit, in service of the inevitable blame game. Democrats in the Senate were never going to let it proceed.

Conservative Republicans welcomed the ensuing confrontation but most of the party did not, because polling shows the public oppose both the wall and a shutdown over it. On Saturday, Senator Lamar Alexander of Tennessee said: “This is a complete failure of negotiations and a success for no one.”

Facebook Twitter Pinterest Jared Kushner, Mick Mulvaney and Mike Pence arrive on Capitol Hill for talks. Photograph: Erik S Lesser/EPA

But Democrats hold a trump card of their own: they will take control of the House in January. Nancy Pelosi of California, due to become speaker, said in a letter to colleagues on Saturday: “Until President Trump can publicly commit to a bipartisan resolution, there will be no agreement before January when the new House Democratic majority will swiftly pass legislation to re-open government.”

Mulvaney threw a dart or two in that direction on Sunday, telling Fox Pelosi was “in that unfortunate position of being beholden to her left wing”. In response, Pelosi spokesman Drew Hamill told Reuters: “House Democrats are united in their opposition to the president’s immoral, expensive and ineffective wall.” He said the White House should “stop the posturing and start serious bipartisan talks”.

Senate leader Chuck Schumer met Pence on Saturday. His spokesman said the two sides remained “very far apart”. On the Senate floor, the New York Democrat said the “Trump shutdown” could end immediately.

“If you want to open the government, you must abandon the wall,” he said.

Democrats have repeatedly said they are open to proposals that do not include the wall, which they say would be costly and ineffective. The party offered this week to keep spending at existing levels, $1.3bn, for fencing and other security measures.

Locked doors, cancelled tours: US national parks suffer amid shutdown Read more

Mulvaney appeared to agree with them on Sunday, saying: “The wall doesn’t solve all of our problems. A border fence does not solve all of our problems.” His remarks – like Corker’s – echoed comments from 2015, unearthed by CNN, in which the then South Carolina congressman said Trump’s wish for a border wall was “absurd and almost childish”.

On those terms, earlier in the week, senators approved a deal to keep the government open. Trump seemed set to approve but then backtracked, apparently under the influence of rightwing media furious at his abandonment of a key campaign pledge, if disregarding his repeated vows that Mexico, not the US taxpayer, would pay.

Meanwhile, the shutdown played out. Around 420,000 essential workers were expected to work unpaid; 380,000 were to stay home without pay. Mulvaney told Fox salaries would be paid on 28 December, adding: “The next pay period that is impacted is 11 January.” Legislation is in train to ensure back pay if needed.

Trump had already declared Monday, Christmas Eve, a federal holiday. Rather than work around the clock, the leaders of the House and the Senate closed shop. But they did not rule out action if a deal were struck.

In New York City, thanks to state funds, the Statue of Liberty was open. But in Washington, another highly symbolic attraction was closed. The National Christmas Tree, near the White House, was dark.