With the partial shutdown in its third week, House Speaker Nancy Pelosi said she intends to begin passing individual bills to reopen agencies in the coming days, starting with the Treasury Department to ensure people receive their tax refunds.
Pelosi and congressional Democrats are at loggerheads with Trump's White House over funding for a southern border wall. Trump wants to erect steel and/or concrete walls over 30 feet tall on more than 200 miles of the border that would include new and replacement barriers.
The White House painted that offer, which Trump floated previously, as an olive branch.
Vice President Mike Pence had planned to meet with congressional leadership staff on January 6, but Trump said any developments related to his $5.6 billion funding request for the wall were likely to happen in the coming days.
A White House official said the meeting included a briefing on border security by Homeland Security Secretary Kirstjen Nielsen.
Top Democrats say President Donald Trump would face legal and political challenges if he tries to declare a national emergency to build the border wall with Mexico. "She went in front of an enthusiastic crowd of her supporters and it was red meat for them".
With his negotiators preparing for the afternoon session, Trump said he didn't expect it to produce a resolution.
He later tweeted that the Pence talks were productive.
Building a wall along the 2,000-mile (3,200-kilometer) US-Mexico border was a central plank in the 2016 election campaign of Trump, who has sought to equate immigrants with crime, drugs and gangs. The aide said no progress was made and no further meetings of the group were scheduled. Democrats familiar with the meeting said the White House position was "untenable". Trump said he wanted to help on the Dreamer issue but preferred to wait for a Supreme Court ruling on it first.
A migrant from Honduras pass a child to her father after he jumped the border fence to get into the USA side to San Diego, Calif., from Tijuana, Mexico. Senate Majority Leader Mitch McConnell has said the Republican-controlled upper chamber won't consider any bills without the president's support.
Americans depend on the federal government for social security retirement supplements, for old-age health insurance, for highway funds, for airport security and for all manners of other services, even as they say they want independence from Washington. About 800,000 government workers are either furloughed or working without pay.
Trump also asserted that he could relate to the plight of the hundreds of thousands of federal workers who aren't receiving paychecks, though he acknowledged they will have to "make adjustments" to deal with the shutdown shortfall.
And when someone asked if that was a threat to Democrats, Trump stated: "I never threaten anybody".
Not all Republicans agree with Trump's insistence on keeping government agencies shuttered until the border debate is resolved.
In an interview with NBC's "Meet the Press", set to air Sunday, Mulvaney argued that the administration was willing to deal.
The second bill would fund the DHS through February so negotiations over the border wall could continue.
Adam Schiff, a California Democrat and incoming chair of the House Intelligence Committee, said the House would need bipartisan support from Republican senators to make it happen. "But if we can do it through a negotiated process, we're giving that a shot".