The disagreement centres on President Donald Trump’s insistence on a bill that funds a US-Mexico border wall.
The previous shutdown, lasting 35 days, was the longest in US history.
The latest impasse has arisen because Democrats want Immigration and Customs Enforcement (ICE) to concentrate on detaining migrants with criminal records instead of those who have overstayed their visas by limiting the number of beds its detention centres have, the New York Times reported.
Democrats were hoping to cap the number of beds at 16,500. That is roughly the number of people detained in the last years of the Obama White House, the Washington Post says.
Negotiators had also been looking at between $1.3 billion and $2 billion in funding for Trump’s proposed border wall, a long way off the $5.7 billion the president has been demanding, reports said.
On Sunday, lead Republican negotiator Senator Richard Shelby told Fox News that he was “not confident we’re going to get there”.
“I’ll say 50-50 we get a deal,” he said, adding: “The spectre of a shutdown is always out there.”
I don’t think the Dems on the Border Committee are being allowed by their leaders to make a deal. They are offering very little money for the desperately needed Border Wall & now, out of the blue, want a cap on convicted violent felons to be held in detention!
However one of the Democratic negotiators, Jon Tester, said he remained hopeful a deal could be reached in time to avoid a new shutdown.
“It is a negotiation. Negotiations seldom go smoothly, the whole time,” he told Fox News Sunday.
On 25 January President Trump agreed to a three-week spending deal to end the shutdown and allow Congress to reach an agreement.
However, he later suggested the talks were a “waste of time”.
Trump had made stopping the influx of undocumented immigrants the focus of his 2016 campaign, and a priority while in office.
His administration had cracked down on immigrants living illegally in the US by aggressively conducting deportations.
The president has backed away from his calls to make Mexico pay for a concrete wall along the border. But during his State of the Union speech on Tuesday he insisted on a “smart, strategic, see-through steel barrier”.
A new shutdown could see federal agencies including the Homeland Security, State, Agriculture and Commerce Departments lose access to money and begin to close down again.
It would affect about 800,000 federal employees, who would go unpaid. During the last shutdown, some employees continued to work unpaid, but many others called in sick.