Trump made his comments during a Christmas photo op in the Oval Office, where he spoke by video teleconference with service members from all five branches of the armed forces who are stationed in Alaska, Bahrain, Guam and Qatar.
About 25 per cent of the government is shut down for the fourth straight day with Congress at a stalemate over Trump's demand for US$5 billion in wall construction money.
"I can't tell you when the government is going to reopen," he said.
"I can tell you it's not going to be open until we have a wall, a fence, whatever they'd like to call it. I'll call it whatever they want. But it's all the same thing. It's a barrier from people pouring into our country."
The President defended his plan to construct a wall along the border with Mexico that he insisted only an Olympic athlete would be able to scale. "If you don't have that, then we're just not opening," Trump said.
All told, about 800,000 of 2.1 million federal workers nationwide - or more than a third - are estimated to be affected in some way by the shutdown. Trump claimed that many of them support the shutdown.
"Many of those workers have said to me, communicated - stay out until you get the funding for the wall," Trump said. "These federal workers want the wall."
But his claim conflicts with the accounts of federal workers' union leaders.
"Federal employees should not have to pay the personal price for all of this dysfunction," said Tony Reardon, president of the National Treasury Employees Union, which represents 150,000 members at 33 federal agencies and departments.
"This shutdown is a travesty. Congress and the White House have not done their fundamental jobs of keeping the government open."
In a survey of 1500 union members, about 85 per cent said they have limited holiday season spending or are planning to do so because of uncertainty about income, Reardon said. He said they include one who said "I have recently lost my wife and now . . .. I have to put off buying her headstone. Breaks my heart."
Another responded, "I can't buy any Christmas presents and am considering returning those I already have. Mortgage companies and utility companies won't wait to get paid," according to Reardon.
The government shutdown comes near the end of what will probably be the worst year for the US stock market since the financial crisis a decade ago. Many sectors are now technically in a "bear market," down more than 20 per cent.
Before the recent reversal in the stock market, which began during the fall, Trump had frequently boasted that stocks were soaring during his tenure, claiming credit for the upswing.
The downturn has caused great uncertainty in the White House as Trump has sought someone to blame, and he has lashed out publicly at Powell.
On Sunday, several news outlets reported that Trump was looking for ways to potentially fire Powell. That spurred Mnuchin to release a statement on Trump's behalf that the President did not believe he had the "right" to fire Powell.
Many economists and analysts on Wall Street say that an attempt to fire the Fed chief would send markets in a tailspin amid uncertainty over the stability of the US central bank, the most important institution in shaping US and global markets.
Trump is angry that Powell is continuing the Fed's long-telegraphed plan to raise its benchmark interest rate and scale back other extraordinary measures intended to support the economy. The Fed's benchmark rate is now 2.25 to 2.5 per cent, higher than the near-zero interest rate the Fed had pursued for years but still historically low.
Trump complains that his predecessor Barack Obama benefited from lower interest rates, spurring economic growth. Others point out that the Fed, which operates independently of the White House, had pursued a low-interest-rate policy because the economy was much weaker than it is today.
"They are raising interest rates too fast, that's my opinion," Trump said today. "President Obama didn't do much of that . . . He had a very low interest rate."
Powell was Mnuchin's recommendation when Trump was seeking a new Fed chair a year ago, and some close to Trump have questioned whether the President will begin to blame the Treasury Secretary, too, for the sell-off. But today he praised Mnuchin, a longtime friend and finance chairman of his 2016 campaign.
Trump tried to place blame for Washington's budget stalemate solely on Democrats, portraying their leaders as disingenuous and claiming that they support the notion of a physical barrier along the border and object to building the wall only because it is Trump's idea.
Trump likened the situation to his May 2017 firing of James Comey as FBI Director, saying "Democrats hated him" until he was fired and then Comey was treated as a hero.
Even as he vowed to keep the government closed to secure wall funding, Trump claimed that much of his envisioned wall was already being built. He said he had awarded a contract "at a great price" for a 185km section of wall in Texas and plans to visit the site in January for "a ground-breaking" ceremony.
Administration officials have provided no details about this construction project or the terms of the contract.
Trump said he hopes to have all portions of the wall completed - either old fencing renovated or new construction - by Election Day 2020, an indication that he sees building the wall chiefly as a political issue tied to his reelection chances.
Trump said that the wall as he envisions it would be a fortress that no normal human could breach.
"Now there may be the case of an Olympic champion who can get over the wall, but for the most part you are not able to do it," he said. "Very high. It's gonna be 30 feet (9m). Much of it is 30 feet high. Some if it's low. But in some areas we have it as high as 30 feet. That's like a three-storey building."
Trump said of the government shutdown, "It's a disgrace what's happening in our country. But other than that, I wish everybody a very Merry Christmas."
