House Democratic leader Nancy Pelosi (D-CA) speaks to reporters with Senate Democratic leader Chuck Schumer (D-NY), Rep. Steny Hoyer (D-MD) and Sen. Dick Durbin (D-IL) following a border security briefing with U.S. President Donald Trump and congressional leadership at the White House in Washington, January 2, 2019.

"A question to the president and Republicans is: Why don't you accept what you have already done to open up the government?" asked Pelosi, who will likely become House speaker on Thursday.

When Democrats take a House majority Thursday, they plan to pass legislation to fund eight federal departments through Sept. 30 and reopen the Department of Homeland Security through Feb. 8 while lawmakers negotiate border money. In remarks after the White House briefing, House Minority Leader Nancy Pelosi and Senate Minority Leader Chuck Schumer challenged Republicans to support the plan — as the Senate GOP did before the shutdown started late last month.

On the 12th day of a partial government shutdown, congressional leaders came no closer to reaching a deal on border security money. Democrats again flatly rejected the president's call for $5 billion to fund a barrier. Meanwhile, Republicans pledged not to take up a Democratic-backed proposal Trump opposes.

Congress and the White House appeared no closer to breaking a government funding stalemate Wednesday as top lawmakers dug in after a White House briefing on President Donald Trump 's proposed border wall.

The shutdown will drag on for now as the new Congress starts and hundreds of thousands of federal workers face missed paychecks. Trump invited congressional leaders back to the White House for negotiations Friday. But based on their comments Wednesday, it may take a while for lawmakers to crack the impasse.

Before the briefing Wednesday, Trump insisted on more than $5 billion in border wall funding — a figure Democrats say they will not accept. The president tried to pin the blame for the closure on Democrats — despite the fact that he said last month that he would be "proud to shut down the government for border security."

"The bottom line is very simple: at our last meeting the president said, 'I am going to shut the government down,'" Schumer said outside the White House on Wednesday. "They are now feeling the heat."

A senior White House official described the meeting as "contentious at times" but added that "it ended on a positive note." The official said the negotiators got a "better understanding" of where their counterparts stand.

Challenges abound in trying to bridge the divide over the wall. Returning to the Capitol on Wednesday, Senate Majority Leader Mitch McConnell stressed that his chamber will not vote on the plan the House aims to pass Thursday because Trump has signaled he will not sign it. He said he did not think "any particular progress was made today."

However, he said "we're hopeful that somehow in the coming days and weeks we'll be able to reach an agreement." The Senate GOP will hold 53 of 100 seats in the chamber starting Thursday, meaning Democrats will need to find a solution the caucus supports.

Democrats previously offered to put $1.6 billion toward border security — but not a wall as Trump proposes — in the spending plan. The White House then floated $2.5 billion in funding, but Democrats rejected that offer. Trump signaled Wednesday that he would not accept anything other than $5 billion for his proposed wall.

In an interview for NBC's "TODAY" show taped before the briefing, Pelosi unequivocally said Democrats would not approve money for the wall.

"We can go through the back and forth. No. How many more times can we say no? Nothing for the wall," the California Democrat said.

Trump's own messaging muddied his push for a wall as talks stalled over the holidays. The president has not made it clear what exactly he wants — calling at various times for a concrete barrier, fencing or a structure made of steel slats.

A tweet Wednesday morning also raised questions about why Trump still demands taxpayer money for the wall. He claimed "Mexico is paying for the wall" through a replacement of the North American Free Trade Agreement. The Trump administration , and Congress still needs to approve the deal for it to take effect.

He also claimed "much of the wall has already been fully renovated or built." Congress has not passed funding for the wall as Trump proposed, but has put money toward replacing existing fencing or building new fences on the border.

Trump tweet

Trump has insisted on funding for the barrier, a core campaign promise that excited supporters at political rallies when he promised Mexico would pay for it. Democrats have flatly opposed the funding, calling a wall inhumane and ineffective.

Disclosure: NBC and CNBC are owned by Comcast's NBCUniversal unit.

— CNBC's Eamon Javers contributed to this report.

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