Trump's demand for wall funding has led to a partial shutdown of the federal government that has so far lasted 16 days.
On Sunday, Mr Trump pledged not to bend in his demand for $5.6 billion to pay for the wall but said the barrier could be made of steel instead of concrete as a potential compromise with Democrats who refuse to fund it. We need something to prevent people from coming into this country illegally, ' Mulvaney said.
A further complication: Trump's long practice of contradicting his aides makes it hard for anyone else to negotiate on his behalf.
"I haven't done it, but I may do it", Trump said.
Trump asserted that he could relate to the plight of the hundreds of thousands of federal workers who aren't getting paid, though he acknowledged they will have to "make adjustments" to deal with the shutdown shortfall. "If that's not evidence of the president's desire to try to resolve this I don't know what is".
"There is a provision in law that says the president can declare an emergency".
Intelligence Committee Chairman Adam Schiff called it a "non-starter". He also threatened to declare a national emergency in order to appropriate funding for the wall, a move that would bypass Congress.
Republican Susan Collins of ME wants the Senate to take up House-passed bills to re-open the government.
Meanwhile, Mick Mulvaney, his acting chief of staff, said the USA president was prepared to take the notion of a concrete wall permanently "off the table" replacing it with a "steel fence" as a concession to Democrats.
In an interview with NBC's "Meet the Press", set to air Sunday, Mulvaney argued that the administration was willing to deal. Collins and others are also up for re-election in 2020 from states including Colorado and North Carolina where views on Trump are mixed.
Mr Trump is trying to convince the Democrats to approve around $5bn (£3.95bn) to build the wall, which he says is vital to stop illegal immigrants and criminals entering the US.
Democratic House Speaker Nancy Pelosi said a priority would be to reopen the Treasury Department and Internal Revenue Service, "an action necessary to make sure working families received their tax refunds on schedule".
"We are now planning a steel barrier rather than concrete (wall)".
"We have a crisis at the border of drugs, of human beings being trafficked all over the world", Trump said.
"In fact, he said he'd keep the government closed for a very long period of time, months or even years", Schumer said. But Democrats have made clear they see a wall as immoral and ineffective and prefer other types of border security funded at already agreed upon levels.
President Trump will travel to the U.S. -Mexico border on Thursday in a bid to highlight border security, as he presses Democrats for wall funding amid the protracted standoff that triggered a partial government shutdown now stretching into its 17th day.
Trump said on Friday he could declare a national emergency to circumvent lawmakers and build the wall, though budget experts said Congress would still need to allocate the funds.
However, such a move could face legal challenges in court.
More discussions are planned for Sunday.