The White House budget director said Sunday that the White House has given a "counter-offer" to the Senate Democratic leader Chuck Schumer, appearing to soften its position on the funding of the president's promised U.S.-Mexico border wall.

"They are 1.3 (billion dollars) yesterday, we are at 5 billion dollars a couple of days ago, and the counter-offer that we gave them yesterday was between those two numbers," White House Office of Management and Budget Director Mick Mulvaney said during an interview on Fox News Sunday.

Mulvaney, also the incoming Acting White House Chief of Staff, said the partial government shutdown, which began midnight Friday due to a budget impasse, will likely extend into the new year, when Democrats take control of the House.

Referring to the "steel slat barrier" fencing that President Donald Trump tweeted on Friday, Mulvaney said "that's what we want to build," adding that "it doesn't have to be a 30-foot tall concrete wall."

Despite that, Schumer reiterated Saturday a physical barrier is off the option. "What we do support -- Democrats and Republicans -- is real, effective border security. But not a wall. The wall is President Trump's bone to the hard right."

"Democrats have always been for smart and effective ways to secure our border," Schumer said. "We are pushing for technology, like drones and sensors, and inspection equipment."

Trump tweeted early Sunday that while "drones and all of the rest are wonderful and lots of fun," only a "good old fashioned wall" is truly effective.

Any spending measure would need 60 votes to clear the Senate, and the Republicans currently hold a slim 51-49 majority.

A few hours after the shutdown began, the U.S. House and Senate convened Saturday noon on Capitol Hill, but failed to break the deadlock over the funding of the border wall, a centerpiece of Trump's 2016 campaign.

The Senate will meet in a pro forma session on Monday and will return on Thursday and consider business if a deal is reached on government funding.

Trump said earlier Saturday that he was negotiating with the Democrats to end the partial government shutdown, but "it could be a long stay" due to ongoing disagreements.

The partial government shutdown closed about a quarter of federal offices, affecting nine Cabinet-level U.S. departments and dozens of agencies, including the departments of Homeland Security, Transportation and Justice.

On Saturday night, the National Christmas Tree outside the White House had its lights off to the disappointment of visitors, with a sign saying the site was closed due to "a lapse in federal appropriations." A similar sign was displayed at the National Archives building.

Office of Management and Budget General Counsel Mark Paoletta said Saturday afternoon that the budget office was striving to make the shutdown "as painless as possible" with a "minimum disruption" for Americans.