Washington (CNN) The highly anticipated confab in the White House Situation Room had just commenced Wednesday when it became clear President Donald Trump and congressional leaders would not emerge with a deal.

Seated at the head of his long board room table, Trump turned to the Senate Democratic Leader Chuck Schumer and began griping about the state of his administration's nominees, whom he complained were languishing without votes, according to an account from two people in the room.

As the men -- separated at the table only by Vice President Mike Pence -- began arguing, Trump reached into his jacket pocket to retrieve a sheet of paper: the "great letter" from North Korean leader Kim Jong Un he had brandished to reporters earlier in the day.

"Read this," he instructed Schumer, before flinging the document over the table in the senator's direction.

It was an unpromising start to a session that only came to illustrate the entrenched positions that have now allowed a partial government shutdown to stretch two weeks . The outburst at the far end of the Situation Room table did little to quell the sense among both White House officials and Democratic aides that the stalemate is nowhere close to being resolved.

As congressional leaders prepare to return again to the White House on Friday, hopes remain slim for a resolution that would end the shutdown, which has left 800,000 federal workers furloughed, 25% of the federal government without funding and many others working without the guarantee of a paycheck.

As Schumer read through the letter that Trump tossed at him, the President instructed his Homeland Security Secretary Kirstjen Nielsen -- beaming into the session on video conference -- to begin her presentation on what the White House has deemed a "crisis" on the border.

Schumer interjected, placing Kim's letter back on the table, and told soon-to-be House Speaker Nancy Pelosi to begin laying out the Democrat's proposals to break the deadlock -- an interruption a Department of Homeland Security spokeswoman later called "incredibly disheartening."

Both sides later emerged to decry the state of talks. And as a new Congress was sworn in, the shutdown entered another day.

The White House has rejected the measure the new Democratic House is expected to pass on Thursday that would reopen the government but not provide the funding for a border wall that Trump has demanded in return.

Quizzed by Schumer in the Situation Room on Wednesday about why he would turn down such a plan, Trump -- after declining to answer twice -- finally conceded that accepting such an offer would make him "look foolish," according to sources familiar with the meeting

It's a message he's received from many of his allies, who have encouraged him to remain firm in his demands for border wall funding, even as the prospects of a bill emerging from Capitol Hill with the funds included grew slim with the swearing-in of a new Democratic House.

On Fox News, the President's cheerleaders are as fixed in their views as ever, accusing Democrats of stonewalling Trump's demands for a border barrier. Even inside the West Wing, the President's senior aides have not counseled him to accept Democratic proposals that would end the shutdown, believing having the border security fight in the headlines is more preferable than the special counsel's Russia investigation or Michael Cohen's prison sentence. Many of Trump's top allies in Congress, such as Sen. Lindsey Graham, have encouraged him to wait for a better offer.

Democrats, meanwhile, are insistent they will not approve any plan that includes funding for the wall.

That's left the key players unsure of how long the shutdown will continue.

"Could be a long time, or it could be quickly," Trump said on Wednesday.

Friday's meeting was not being viewed optimistically by most aides involved in the stalled negotiations. One White House official suggested Pelosi would be better positioned to deal now that she's been sworn in as House speaker on Thursday. But there is little to indicate she is willing to back down from her firm stance opposing any border wall funding.

"We can go through the back and forth," Pelosi said in an interview with NBC News aired on Thursday. "No. How many more times can we say no? Nothing for the wall."

White House aides also say privately they are unsure of what Trump will accept from Democrats after he publicly rejected a figure his own vice president had floated to Democrats in the days before the shutdown.

"No, not $2.5 billion, no -- we're asking for $5.6," Trump said in a meeting of his Cabinet on Wednesday. Later, in the Situation Room, Trump did appear willing to accept a lower figure, according to people familiar with the meeting. But Schumer and Pelosi remained firm they would not go above the $1.3 billion that is included for border security in the measures that were due to be voted on in the House on Thursday.

Without clarity on what number Trump would accept, neither side was anticipating a breakthrough at Friday's meeting, which will again convene congressional leaders with the President at the White House.

Initially expected to travel this week to Florida for a policeman's benefit gala at his Mar-a-Lago estate, Trump will instead spend another weekend at the White House.