The shutdown, affecting about one-quarter of the federal government including 800,000 or so workers, began when funding for several agencies expired.
Meanwhile, neither side appeared ready to budge off their negotiating positions.
But the senator from SC said there would never be a government spending deal that doesn't include money for a wall or other physical barriers on the U.S. -Mexico border.
Donald Trump's outgoing chief of staff on Sunday admitted the president's planned wall along the U.S. and Mexico border would be more like a fence, as he gave his final interview before leaving office.
After canceling a vacation to his private Florida club, Trump spent the weekend at the White House. "But we left a solid concrete wall early on in the administration, when we asked people what they needed and where they needed it".
Mr Kelly, who will leave his post on Wednesday after 17 months in the job, made the admission in an extensive interview with the Los Angeles Times.
"It is with them", she said, explaining that Mr Trump is not reaching out to Democrats. "Hard to believe there was a Congress & President who would approve!" "It is with them", she told Fox News. "I am in the White House waiting for the Democrats to come on over and make a deal on Border Security", Trump tweeted earlier.
In the Los Angeles Times interview, Kelly conceded that Trump often pressed against the legal boundaries of his role, as the former secretary of state, Rex W. Tillerson, said recently.
"At this point, it's clear the White House doesn't know what they want when it comes to border security", said Justin Goodman, Mr Schumer's spokesman.
As a partial shutdown of the U.S. government entered its eighth day, with no quick end in sight, the Republican president was in Washington, sending out tweets attacking Democrats and talking up possibly improved relations with China.
Court: Justice Ginsburg up and working after surgery
The projected date of Ginsburg's discharge from the hospital and her return to the court are unknown at this time. Despite this, Trump took to Twitter to wish her a speedy recovery, as did her legions of supporters.
"One of the reasons why it's so difficult to keep people from coming - obviously it'd be preferable for them to stay in their own homeland but it's difficult to do sometimes, where they live - is a insane, oftentimes conflicting series of loopholes in the law in the United States that makes it extremely hard to turn people around and send them home", Kelly said.
"To my Democratic friends, there will never be a deal without wall funding", Graham said on CNN before the meeting with the president.
The House and Senate gaveled in and out of session in under five minutes on Thursday.
"The president was upbeat, he was in a very good mood, and I think he's receptive to making a deal", Graham said, adding that Trump found the potential Dreamer concession "interesting".
He said the hardline approach to illegal immigration that had seen children separated from their parents at the border was the brainchild of ex-attorney-general Jeff Sessions, who "surprised us" by announcing the move.
For instance, members of the US Coast Guard will receive their final paychecks of the year on Monday, the service said in a statement on its website on Friday after previously warning that payments would be delayed due to the shutdown. He's in the - he's in the White House.
As he called for Democrats to negotiate, Trump brushed off criticism that his administration bore any responsibility for the recent deaths of two migrant children in Border Patrol custody.
"Any deaths of children or others at the Border are 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".
Mr Trump is sticking with his demand for money to build a border wall with Mexico, and Democrats, who take control of the House on January 3, are refusing to give him what he wants.
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