Without a budget all but the essential services have shut down and about 800,000 federal employees are on furlough or temporary layoff and have not been paid, while Trump and his adversaries trade charges without meaningful negotiations for the $5.6 billion demand for the border barrier.
Trump, who had made the border wall a key element of his election campaign, is uncompromising and both he and the Democrats are waiting to see who blinks first.
The essential government services manned by workers who haven't received their pay since January 11 are beginning to show signs of fraying.
Employees of Transportation Safety Administration (TSA) that inspects airline passengers have started calling in sick in large numbers leading to shutdown of some checkpoints at some airports and long delays at the others.
The New York Times has published an op-ed calling for the TSA employees to strike.
Without their paychecks, employees working and those who are not have said they are unable to meet their essential living expenses.
About 36,000 employees of the Internal Revenue Service (IRS) were asked on Tuesday to return to their jobs to prepare to process the the avalanche of tax returns over the next three months.
The Federal Aviation Agency also called back over 3,000 aviation safety workers to ensure airline safety.
On Monday, the Food and Drug Administration had begun bringing in hundreds of staff, mainly inspectors, to ensure safety in the food system.
The employees who work through the shutdown will eventually receive their paychecks when a budget is in place.
Last week, the House of Representatives voted a bill to fund the Treasury Department and the IRS, but the Senate has not taken it up.
Shutting down the government by withholding a budget is a political tactic that has been used in the past.
The longest shutdown previously was for 21 days in 1995-96 when party roles were reversed during a standoff between Republicans led by Speaker Newt Gingrich and Democrat President Bill Clinton.
Democrats boycotted a meeting Trump convened at the White House on Tuesday while Nancy Pelosi, who took over as speaker after the Democrats took control of the House in the November mid-term elections, and Chuck Schumer, the leader of the Democrats in the Senate, have refused to compromise.
Schumer blamed Trump for the standoff saying that he banged on the table and walked out during their last meeting.
Three Republican senators have suggested a compromise of funding the government for three weeks during which the two sides can try to work out a deal on the border wall.
Media reports have said that some Republicans - as well as some Democrats - have grown uneasy over the prolonged shutdown and are beginning to ask their leaderships to compromise.
The only compromise that Trump has offered is setting up a steel fence instead of a wall along the border to assuage those opposing a wall.
He has also threatened to impose a national emergency and bypass the Congress to start constructing the wall.
Trump has asserted that there is an emergency due to illegal immigration and drug smuggling, which the Democrats deny.
A caravan of about 7,000 Central Americans, who marched to the border with Mexico, is camped out at the frontier hoping to make its way into the US. Some who tried to crash the border were repelled by border guards teargas.
Perhaps, encouraged by the standoff, another caravan is reported to have started from Guatemala this month.
Last week, Trump cited the case of an Indian-American police corporal, Ronil Singh, who was killed by an illegal immigration when he made a televised appeal to the nation last week to pressure the Democrats to compromise on the border wall.
Trump had said during his campaign that he would make Mexico pay for it, an obvious hyperbole that will not work, requiring him now to seek tax funding to fund it so he can keep his word with his base.
Schumer and some Democrats, who had supported a border barrier in the past now oppose it.
Some Democrats are heading towards a virtual open-border policy.
Senator Kirsten Gillibrand, who plans to run for president, and newly-elected Representative Alexandria Ocasio-Cortez have gone to the extent of calling for abolishing the Immigration and Customs Enforcement agency, which implements immigration and anti-smuggline laws within the US.
(Arul Louis can be reached at firstname.lastname@example.org and followed on Twitter @arulouis)