Mulvaney also echoed a claim from President Trump and the White House that Mexico will "pay" for the wall through cost savings to the U.S. from the new U.S.-Mexico-Canada trade agreement, or the USMCA.
On Tuesday, ABC News Senior National Correspondent Terry Moran pressed Sarah Sanders, the White House press secretary, on that point.
Sanders said "we're not asking American taxpayers" for money for the wall and added that "the president has been clear that the USMCA deal would provide additional revenue through the deal that would show that Mexico has paid for the wall."
Moran pushed back, saying: "But trade benefits go to private citizens. They don't go to the United States Treasury."
Mulvaney said that through the new agreement, "American workers are going to do better, the government is going to do better, and you could make the argument that Mexico is paying for it in that fashion."
At 12 a.m. Saturday, the government partially shut down after the White House and Congress failed to reach a funding deal, with President Trump saying he wouldn't sign a bill that didn't include $5 billion for the construction of a wall along the U.S.-Mexico border, one of his core campaign promises.
Sen. Chuck Schumer, the Senate minority leader, and Rep. Nancy Pelosi, the incoming speaker of the house, have said that Democrats will not agree to any appropriations bill that includes money for a border wall. An NPR/PBS NewsHour/Marist poll released earlier this month showed that 57 percent of Americans believed the president should compromise on border wall funding to keep the government open.
About 75 percent of the federal government has already been funded, but key agencies like the State Department, Justice Department and Department of Homeland Security shut down for all except nonessential employees when no agreement was reached.
Congress has adjourned for the holidays without a deal between lawmakers and the administration. The Senate reconvenes on Thursday, but Senate Majority Leader Mitch McConnell left open the possibility of convening earlier if a deal is struck. House Majority Whip Steve Scalise's office told ABC News there would not be votes in the House before Thursday.
Mulvaney said Sunday that the negotiation "seems like it's going in the wrong direction." He said Democrats were offering $1.6 billion for border security a couple weeks ago, a figure that has now dropped to $1.3 billion.
“[I’m] not going to tell you what our bottom line is in the negotiation, but it’s a fair question as to what we would accept,” Mulvaney said when asked whether President Trump would agree to a deal with no wall funding. “No, the president’s not going to not accept money for a border wall. Now, what one people call a wall and another person might call a fence.”
"The president has made it very clear, however, that he is willing to discuss a larger immigration solution," Mulvaney said. "The wall does not solve all of our problems. A border fence does not solve all of our problems, because so many of our laws are broken."
Trump seemed to counter Mulvaney's claim just minutes after he addressed the issue on "This Week," tweeting that "...it is only a good old fashioned Wall that works!"
President Trump is blaming Democrats in Congress for the shutdown despite claiming last week in a meeting with Schumer and Pelosi that he would take responsibility for a shutdown.
On Dec. 11, Trump told the Democratic leaders that he would "take the mantle" and be "proud to shut down the government for border security."
But on Friday, the president tweeted "The Democrats now own the shutdown!"