Donald Trump and top congressional leaders failed to resolve a partial government shutdown that has stretched well into a second week as the president refused to back off from his demands for billions of dollars for a long-promised wall along the southern U.S. border with Mexico.

Democratic and Republican leaders from both chambers were invited to the the White House's Situation Room, the inner sanctum for classified meetings, on Wednesday for a "border security briefing," The Guardian reported.

During the meeting Trump asked Department of Homeland Security officials to "make a plea" for his wall.

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At a cabinet meeting prior to the briefing, Trump warned that parts of the government would could remain closed for a "a long time" without a deal.

"We're asking the president to open up government," Nancy Pelosi, who is expected to assume the speakership of the House of Representatives on Thursday, said after the briefing with Trump. "We are giving him a Republican path to do that. Why would he not do it?"

The shutdown was triggered by Trump's demand that Congress allocate more than $5 billion in taxpayer money to build a wall along the 2,000-mile border between the U.S. and Mexico – a concession Democrats refuse to make.

The shutdown, which entered its 12th day on Wednesday, has affected nearly 800,000 federal workers. The incoming House Republican leader, Kevin McCarthy, said Trump invited leaders back to the White House on Friday.

Pelosi said the Democrats will still vote on Thursday, as Congress starts its new session with the Democratic party in the majority in the House after victories in the midterm elections last November. Democrats intend to introduce a pair of funding bills that would end the shutdown, but without money for a border wall. The proposal includes $1.3 billion for border security measures that can be used to repair and replace fencing and existing portions of the wall.