Trump is demanding at least $5.6 billion for the wall.
A Reuters/Ipsos opinion poll suggested 50 percent blamed the president for the shutdown, while 32 percent saw the Democratic Party as culpable.
"Look, if Harry Truman couldn't nationalize the steel industry during wartime, this President doesn't have the power to declare an emergency and build a multibillion dollar wall on the border".
Trump has threatened to declare a national emergency to commence building, but that plan of action is likely to face numerous legal challenges and questions of where the money will come from in the midst of a shutdown.
Most recently, Sanders claimed on Sunday in an interview on the usually Trump-friendly Fox News channel that "nearly 4,000 known or suspected terrorists come into our country illegally".
"We've been in touch with a lot of people, and I informed my folks to say that we'll build a steel barrier".
Also, on "This Week", Rep. Liz Cheney, R-Wyoming, a member of the House Armed Services Committee, said of the possible national emergency declaration that "nobody wants to invoke it", but the border wall issue is "completely captive" to congressional Democrats.
House Democrats plan to pass a series of bills this week to reopen government, breaking up legislation they have already approved in a bid to get Republicans to agree to reopen certain agencies, Hoyer said on "Meet the Press".
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Trump skipped a planned trip to Florida to stay in the USA capital during the Christmas and New Year's holiday after large chunks of the federal government were shuttered on December 22.
But Mick Mulvaney, the acting White House chief of staff, said that while he thought the shutdown was "going to drag on a lot longer", Trump's shift in wall materials could provide a semantic opening to advance the talks. "The president has asked every single Cabinet secretary and the Office of Management and Budget to go out and find money that can be used legally to guard the southern border, which is exactly what we're going to do".
"We hope they take us up on it", he said. Speaker of the House Nancy Pelosi (D-CA) has stated openly that she will not give more than $1 for the wall; Senate Minority Leader Chuck Schumer (D-NY) and others have stated that they believe a wall somehow violates American values.
Asked about the ability of government workers to pay their bills, President Trump said Sunday that he expected federal workers would be able to "adjust" financially, as he expressed confidence that they supported his drive for border security.
While Trump had expressed little hope for the talks Sunday, he indicated that the coming days would offer a chance to show that resolve. After bowing to his base and refusing to compromise with Democrats, Schiff said, Trump needs to "figure out how he unpaints himself from that corner".
"They are essentially the same funding bills that the Republican Senate wrote and approved by a 92-6 margin during the last Congress", Hoyer said in a statement Sunday.
But Senate Majority Leader Mitch McConnell, whose chamber passed almost identical bills previous year, refuses to allow the Senate to vote on the measures because Trump has threatened he would veto them.
"Are you recalibrating your assessment of how you can work with this president?"