"An all concrete Wall was NEVER ABANDONED, as has been reported by the media", the president said. "I am in the White House waiting for the Democrats to come on over and make a deal on Border Security", Trump tweeted earlier.
Speaking of the wall, Gen. Kelly said that after consultation with border patrol agents it became clear that a continuous, opaque structure would not be suitable for stemming the flow of illegal immigrants. "They can't. If we had a Wall, they wouldn't even try!" he said.
He said she last spoke to Trump on December 11, by phone hours after leaving the White House.
He also took pride that he kept politics out of the decision-making process at the White House, an issue that Trump's allies repeatedly expressed frustration about the new chief of staff.
Trump wants $5 billion as a down payment on the barrier that could cost more than $20 billion, but Democrats have only agreed to $1.6 billion to improve border security, but no wall money. Trump has faced mounting conservative criticism for his inability to deliver on his core campaign promise.
The measures include a mix of short-term and long-term funding plans to re-open the government agencies that have been closed for 10 days, after a dispute over funding the construction of a wall on the U.S. -Mexico border.
The House is preparing to vote on the package Thursday when the new Congress convenes.
"Now we have to do it the hard way, with a Shutdown", Trump wrote.
When asked about security concerns at the border, Mr Kelly admitted America had an "immigration problem".
Earlier Sunday, Graham told Bash on the same program that he hoped to end the shutdown by offering Democrats incentives to get them to vote for wall funding.
Democrats have said they believe investments in border security are necessary, but the idea of building a physical barrier on the border with Mexico would be a waste of money.
In other words, Democrats will seek to detach the border wall fight with the president from other appropriations and kick the final reckoning on border security into February.
House Democrats have come up with a plan to reopen the government when they take control of the chamber on Thursday - but without the $5 billion President Trump is demanding for a border wall. Democrats in the Senate promptly threatened to filibuster any funding measure that provided for a border wall, and the two sides have been at an impasse ever since, with no end in sight. Trump disputes Kelly's comments on the wall.
"What happened was Jeff Sessions, he was the one that instituted the zero-tolerance process on the border that resulted in both people being detained and the family separation", Kelly said.
"I think the President's eventually going to win his court case regarding DACA and the President has put on the table several times relief for the DACA population".
Trump has consistently painted the asylum seekers and economic migrants in outlandish terms, raising the spectre of rapists, gang members and people with infectious diseases roaming freely across the border. But Democrats are wary of entering into any immigration deal with Trump after previous talks linking the fate of dreamers to wall funding collapsed.
The shutdown is forcing hundreds of thousands of federal workers and contractors to stay home or work without pay.