As the disruption in federal services and public employees' pay appeared set to continue into the new year, there were no signs of any substantive negotiation between the blame-trading parties.
Trump is holding out for billions in federal funds for a border wall between the US and Mexico, which Democrats have said they were intent on blocking.
Trump tweeted Saturday that he was "in the White House waiting for the Democrats to come on over and make a deal on Border Security."
But there has been little direct contact between the sides during the stalemate, and Trump did not ask Republicans, who hold a monopoly on power in Washington for another five days, to keep Congress in session.
As he called for Democrats to negotiate on the wall, Trump brushed off criticism that his administration bore any responsibility for the recent deaths of two migrant children in Border Patrol custody.
Trump claimed the deaths were "strictly the fault of the Democrats and their pathetic immigration policies that allow people to make the long trek thinking they can enter our country illegally."
His comments on Twitter came as his Homeland Security secretary met with medical professionals and ordered policy changes meant to better protect children detained at the border.
Democrats are vowing to pass legislation restoring the government as soon as they take control of the House on Thursday, but that won't accomplish anything unless Trump and the Republican-controlled Senate go along with it.
Talks have been at a stalemate for more than a week, after Democrats said the White House offered to accept USD 2.5 billion for border security last Saturday.
Senate Democratic leader Chuck Schumer told Vice President Mike Pence that it wasn't acceptable, nor was it guaranteed that Trump, under intense pressure from his conservative base to fulfill his signature campaign promise, would settle for that amount.
Conway claimed Sunday that "the president has already compromised" by dropping his request for the wall from USD 25 billion, and she called on Democrats to return to the negotiating table.
But Conway indicated Trump has moved off his demand for a physical wall along parts of the border, as he promised during his 2016 campaign, calling discussion of a wall "is a silly semantic argument."
"There may be a wall in some places, there may be steel slats, there may be technological enhancements," Conway said. "But only saying 'wall or no wall' is being very disingenuous and turning a complete blind eye to what is a crisis at the border."
Trump has remained out of the public eye since returning to the White House early Thursday from a 29-hour visit to US troops in Iraq, instead taking to Twitter to attack Democrats. He also moved to defend himself from criticism that he couldn't deliver on the wall while the GOP controlled both the House and Senate.
"For those that naively ask why didn't the Republicans get approval to build the Wall over the last year, it is because IN THE SENATE WE NEED 10 DEMOCRAT VOTES, and they will gives us "NONE" for Border Security!," he tweeted. "Now we have to do it the hard way, with a Shutdown."
Meanwhile, the effects to the public of the impasse grew as the Environmental Protection Agency, which had the money to function a week longer than some agencies, implemented its shutdown plan at midnight Friday night.
EPA spokeswoman Molly Block said many of the agency's 14,000 employees were being furloughed, while disaster-response teams and certain other employees deemed essential would stay on the job.
That includes workers needed for preventing immediate public health threats at more than 800 Superfund hazardous-waste sites.
Also running short on money: the Smithsonian Institution, which said its museums, art galleries and zoo in the capital will close starting mid-week if the partial shutdown drags on.
But federal flood insurance policies will continue to be issued and renewed, in a reversal prompted by pressure from lawmakers, said Republican Sen. Marco Rubio of Florida.
Trump appeared no closer to securing money for his signature border wall, which he vowed during the campaign that he would make Mexico pay for. He's failed to do so. Now Democratic leaders are adamant that they will not authorize money for the project, calling it wasteful and ineffective. They show no signs of bending, either.
The shutdown is forcing hundreds of thousands of federal workers and contractors to stay home or work without pay.
The White House has not directly engaged in weeks with the House Democratic leader Nancy Pelosi, who has all but locked up the support she needs to win the speaker's gavel after the new Congress convenes on Thursday.
Pelosi has vowed to pass legislation to reopen the nine shuttered departments and dozens of agencies now hit by the partial shutdown as soon as she takes the gavel.
Pelosi spokesman Drew Hammill added that Democrats are united against the wall and won't seriously consider any White House offer unless Trump backs it publicly because he "has changed his position so many times".