The parks service is one of the many federal institutions suffering from the funding suspension.
The government partially shut down at midnight Friday after Congress and the White House failed to agree on a spending package, the third shutdown this year.
"It's very possible that the shutdown will go beyond the 28th and into the new Congress", White House Budget Director Mick Mulvaney said on "Fox News Sunday".
The US president said border security remained at the centre of discussions after he demanded $5bn (£4bn) to fund the construction of a wall along the US border with Mexico.
Last week he said he would be "proud" to close down the government, and last Friday said he was "totally prepared for a very long" closure. Now, three days into the shutdown, his budget chief says it could drag into the New Year.
A gate leading to public restrooms is locked at the Ellipse visitor centre near the National Christmas Tree that is now closed to the public due to the partial shutdown.
Essentially, it all boils down to Trump's much-vaunted US-Mexico border wall that Trump says is essential to stop illegal immigration.
Earlier this week, he said: "This is our only chance that we'll ever have, in my opinion, because of the world and the way it breaks out, to get great border security". "This is politics, not policy".
Byungsuk Ko / EyeEm via Getty Images The Statue of Liberty will remain open after an emergency injection of state funds. Others deemed essential, including Transportation Security Administration officers screening passengers during the holiday crush, are working unpaid. Another 380,000 were to be furloughed, meaning they will not report to work but would be paid later. The Senate had already passed legislation ensuring that workers will receive back pay, and the House was likely to follow suit.
A Wonderful Life with Trump? President calls SNL 'Democratic spin machine'
A man laughs at another man in a Donald Trump costume at the Fans Village in Alexandra Palace on December 16 in London, England. Earlier, he took specific aim at Baldwin's impersonation of him, calling it an "agony for those forced to watch".
"It's Christmas Eve and President Trump is plunging the country into chaos", the two said in a joint statement.
But they did not rule out action if a deal were struck. "The next scheduled session day will be on the 27th of December", Majority Leader Mitch McConnell said on the Senate floor. Absent were GOP leaders or Democrats who would be needed for a deal.
"We moved off of the five and we hope they move up from their 1.3", Mulvaney said less than a day after a senior administration official insisted that Congress would have to cave into Trump's demand for the shutdown to end, highlighting Trump's unpredictable negotiating style.
Trump had savored the prospect of a shutdown over the wall.
The total cost at that point, including personnel, was approximately $2.3 billion, the government watchdog said.
"Instead of bringing certainty into people's lives, he's continuing the #TrumpShutdown just to please right-wing radio and TV hosts", Schumer tweeted, referring to Rush Limbaugh and Ann Coulter, among others, who ripped him for being unable to get funding for his long-promised border wall.
"When everybody leaves town, it takes the pressure off for a deal", said New York Representative Lee Zeldin, a Republican, one of the few lawmakers lingering in the Capitol on Sunday.
Democrats have said they are open to budget proposals that do not include the wall, which they say would be costly and ineffective.
Those being furloughed included almost everyone at NASA and 52,000 workers at the Internal Revenue Service. Also still functioning were the Federal Bureau of Investigation, the Border Patrol and the Coast Guard. A drawdown of troops in Afghanistan also appeared to be in the works.