There's no requirement that this government be shut down while we deliberate the future of any barrier, whether it's a fence or a wall," said Sen. Dick Durbin, D-Ill., on CBS' "Face the Nation."

Senior congressional aides and top White House officials, including Vice President Mike Pence, met again Sunday in the nation's capital after hours of talks the previous day did not lead to a compromise. But no deal was made, and no further meetings for the group are scheduled, according to a Democratic official familiar with the meeting.

On Saturday, Democrats asked the White House for a budget justification for the requested funds, which were not included in the administration’s 2019 budget. That justification was not completed Sunday, the official said.

Pence's office described Saturday's meeting at the Eisenhower Executive Office Building, next door to the White House, as "productive" with Democratic sources saying those in the room urged Trump to reopen the government and back off his request for more than $5 billion to construct a wall along the southern U.S. border.

Mulvaney said he thought the meeting was "disappointing" and that Democrats were not looking for a solution "because they think that they are winning this battle politically."

The president admitted on Twitter after the Saturday meeting that "not much headway" was made during the talks but again pressed for border security, a sign neither side is wavering in the bickering that has left thousands of federal employees working with no pay.