A congressional deal seemed to stall even after Mulvaney convened a bipartisan group of lawmakers on Friday.
The sticking point is over the number and objective of immigration detention beds, an issue that would typically be regarded as a side-note to the broader talks about the amount of funding for border security. Trump has called for $5.7 billion, but lawmakers were trying to find a number between $1.3 billion and $2 billion that would be acceptable to both sides. If an agreement on funding the government isn't reached by Friday at midnight, the government could partially shut down again, just three weeks after the longest USA government shutdown in history.
"I'm not confident we're going to get there, I'm hoping we will get there", Shelby said regarding the prospect of reaching a deal.
Republicans favor tough enforcement of immigration laws and have little interest in easing them if Democrats refuse to fund the Mexican border wall. Republicans have said violent criminals caught inside the United States shouldn't count toward the cap.
White House chief of staff Mick Mulvaney, meanwhile, said a government shutdown is "still technically on the table". "There'll be some things that simply we couldn't agree to".
That is roughly the number of people detained in the last years of the Obama White House, the Washington Post says.
Mulvaney added however that it may not be necessary for Trump to secure the funds through an emergency declaration, but that such a move would be legal.
"We are not interested in our committee in whether he's a tax cheat or not worth what he says he is", Rep. Adam B. Schiff (D-Calif.) said in an appearance on the NBC's "Meet the Press".
The White House said Sunday it can not rule out another government shutdown as it gets mixed signals from Democrats and Republicans racing a Friday deadline to fund President Trump's border-wall demands and keep federal operations humming.
Predictably each side blamed the other for the stall in negotiations. Details about where the fencing would go and a Democratic request to eliminate previously funded fencing in the National Butterfly Center, a conservation area close to the border in Mission, Texas, were still being negotiated.
The president once again accused the House Democrats of refusing to provide enough money for his cherished wall on the Mexico border - an issue that has been a stumbling block on the border security negotiations from the very beginning.
The mood among bargainers has soured, according to people familiar with the negotiations.
The White House and many Republicans want to push the amount that would be spent for building physical barriers to $US2 billion ($A2.8 billion) or higher.
A Democratic aide briefed on talks said the discussions "broke down" on Saturday after the Senate GOP rejected their position on limits for Immigration and Customs Enforcement detention beds in the interior of the country.