The US government shutdown that has halted pay for hundreds of thousands of federal workers began its third week with no end in sight, as Vice-President Mike Pence, top White House officials and senior congressional aides met for more than two hours without reaching a deal to reopen the government.

Inside the meeting at the Eisenhower Executive Office Building, Pence refused to budge from the more than US$5 billion President Donald Trump has demanded from Congress to pay for a portion of his promised wall along the US-Mexico border, according to two Democratic officials.

The standoff - which has heavily affected national parks and other operations and threatens to halt payments as varied as food stamps and tax refunds - has made Trump's unrealised wall the linchpin of his presidency as he seeks to make good on a signature campaign promise.

The federal agency tasked with guaranteeing US airport security acknowledged an increase in the number of its employees calling off work during the partial government shutdown.

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Employees of the Transportation Security Administration are expected to work without pay during the shutdown because their jobs are considered essential. The TSA said that call outs that began over the holiday period have increased.

Administration officials have acknowledged that they were not prepared for the potential consequences of an extended shutdown and Trump's decision to demand wall funding. Democrats are standing firm on offering no taxpayer money for the project, which Trump had asserted would be funded by Mexico.

Pence did not have the President's blessing to float new or specific numbers as he did last month in a meeting with Senate Minority Leader Charles Schumer, according to two Trump aides. That meant few specifics were actually discussed, as Democratic staffers repeatedly pushed the Administration to reopen the federal government and negotiate differences over the border after the shutdown ends.

But the Administration - represented by Pence, Homeland Security Secretary Kirstjen Nielsen, acting White House chief of staff Mick Mulvaney and senior adviser Jared Kushner - refused, according to multiple officials.

"Not much headway made today," Trump tweeted. "Second meeting set for tomorrow. After so many decades, must finally and permanently fix the problems on the Southern Border!"

Trump spent much of the day on the phone with allies, talking through his positioning on the shutdown. Trump told reporters on Saturday that he wants to reopen government but is prepared to maintain the shutdown for weeks or even years.

Democratic staffers asked the White House to lay out in formal detail the Administration's funding request for the border - including its specific security requests, what the money would be used for, and what in the Homeland Security budget the Administration would cut to make the numbers work, people familiar with the meeting said. The White House plans to provide those figures before the group meets again today.

House Speaker Nancy Pelosi said that the House will begin passing separate bills to reopen the government this week, starting with the funding bill that covers the Treasury Department "so that the American people can receive their tax refunds on schedule. The senseless uncertainty and chaos of the Trump Shutdown must end, now".

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Mulvaney said that Trump would "take a concrete wall off the table" in negotiations with Democratic leaders if that would help end the shutdown.

"If he has to give up a concrete wall, replace it with a steel fence to do that so that Democrats can say, 'See? He's not building a wall anymore,' that should help us move in the right direction," Mulvaney said.

Trump is annoyed by news reports about the negotiations that make it seem that he is backing away from his demands and wants to avoid stories about new numbers for wall funding being discussed, aides said.

Meanwhile, House lawmakers are planning to roll out a measure on Wednesday that would require universal background checks for nearly all firearms purchases, as House Democrats attempt to make good on their pledge to address gun control with their new majority.

The proposal, headlined by Pelosi and Gun Violence Prevention Task Force Chairman Mike Thompson, is a bipartisan measure, with Peter King serving as the lead Republican co-sponsor. It will seek to impose universal background checks for the purchase and transfer of firearms, with some exceptions for hunting and family, but will not address an assault weapons ban.

Former congresswoman Gabby Giffords, herself a shooting victim, plans to be in Washington.

- Additional reporting AP