Democrats have criticised plans like the US$400 million for broad use as "slush funds," and the offer left the two sides far apart.
Democrats have refused any new money for a wall along the US-Mexico border, which Trump during his campaign promised would be paid for by Mexico.
"There is frankly no path towards him getting US$5 billion in American taxpayer money to meet his campaign promise of a big, beautiful wall with Mexico," said Senator Chris Coons, D, on CBS.
Coons suggested Trump accept US$1.3 billion for border security - the existing level which doesn't include any new money for building a border wall.
"There is a path towards our responsibly appropriating about US$1.3 billion for border security; I'll remind you, the Administration hasn't spent yet the US$1.3 billion we appropriated this year. I think the President simply needs to hear, 'yes,' and we all need to move forward."
Shortly before the shutdown began, the Senate agreed to a bipartisan deal that would keep the government open through to February 8 but deny Trump any new wall money, keeping the Department of Homeland Security's border security funding at the current level of US$1.3 billion.
Lawmakers were initially hopeful Trump would take that deal, especially after he suggested the wall could still be built by the military. But that strategy faced high legal hurdles, and after a backlash some of Congress' most conservative members and typically Trump-friendly media outlets and pundits, the President torpedoed the deal and resumed his demand for more than US$5 billion in wall funding.
Today's statements further entrench both sides in an impasse that led large portions of the government to shut down at the weekend.
But Mulvaney also acknowledged that Trump's approach to the presidency played a role in closing federal agencies and sending workers home just before Christmas.
"This is what Washington looks like when you have a president who refuses to sort of go along to get along," Mulvaney said.
The House and Senate have been sent home until Friday NZT, after several hours of fitful negotiations at the Capitol yielded no result. But Mulvaney indicated little optimism that there would be any deal to reopen the government before the new Congress convenes on January 4.
The likely incoming House speaker, Congresswoman Nancy Pelosi, D, has already said she intends to pass a spending bill to reopen the government as soon as she takes control of the House majority. It would provide US$1.3 billion for border security, a level Democrats have endorsed and that would contain no new funds for building a wall.
Mulvaney said that the White House made an offer between US$1.3 billion and US$5 billion, and "the ball is in their court".
Democrats have firmly opposed any new money for Trump's border wall. Pelosi says the wall is "immoral".
Mulvaney played down the impacts of a shutdown, noting that the federal government is effectively closed until Thursday anyway because of the Christmas holiday. He said that pay will go out on December 28, the next federal pay period. "I want everybody to understand no one is working without getting paid," Mulvaney said.
The next pay period affected is January 11.
Some 400,000 federal workers are expected to be at home under the shutdown that affects about 25 per cent of the government that Congress funds - including the Homeland Security Department, and the Justice, Interior, Agriculture and Housing departments, among others.
The rest of the government - including the Pentagon - have already been funded through to September under spending bills Congress passed earlier in the year and the president signed.
The government also went into brief partial shutdowns in January and February, making this the third partial government shutdown of 2018.
When ABC's Jonathan Karl reminded Mulvaney of Trump's campaign promise that Mexico would pay for the wall, Mulvaney said the technical process cannot work so easily. He tried to argue that the renegotiated trade deal among the United States, Mexico and Canada could generate revenues to the US Treasury.
"You could make an argument Mexico is paying for it in that fashion," Mulvaney said, without offering specifics. He added, "We really think we're in a good place in terms of getting the wall built and also getting Mexico to participate in our border security."
There is no mechanism for direct payments from Mexico's government to the US government for a wall in the trade agreement.
Trump continued his public press for the wall, writing on Twitter that a "Wall or Barrier" is the only effective way to stop undocumented immigration - a claim many border experts have contested.
"Drones and all of the rest are wonderful and lots of fun, but it is only a good old fashioned Wall that works!" he wrote.
