Vice President Mike Pence, speaking after a border-security briefing at the White House on Monday, told reporters the White House Counsel's office has been looking into the legality of President Trump declaring a state of emergency to authorize the construction of a border wall without approval or funding from Congress.

Trump repeatedly has floated that possibility in recent days, saying on Sunday that he "may declare a national emergency dependent on what's going to happen over the next few days." The statement marked the clearest indication yet that the White House actively was pursuing that approach.

“There has been a precipitous rise in illegal immigration at our southern border, particularly unaccompanied children and families," Pence said. "Democrats don’t dispute the facts of the problem at the border. That is the foundation for finding agreement on solutions.”

Pence has held two meetings with congressional officials in the past week. On Sunday, a Democratic official told Fox News that Republicans were to blame for the meeting "starting 45 minutes late" and for failing to provide a "full budget justification" for their $5.7 billion request for wall funding. Trump, however, called the meeting "productive," and a House GOP leadership aide told Fox News that Democrats had received an exhaustive cost "breakdown" as requested.

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Pence said Monday that he provided Democratic Sens. Patrick Leahy of Vermont and Chuck Schumer of New York with a written offer to end the shutdown, but it was refused. The vice president did not confirm that he had asked for $2.5 billion for the wall -- which would represent a significant concession from the administration's previously stated goal of more than $5 billion -- but smiled when asked if that was, in fact, the number.

Trump, on Sunday, formally committed to a steel wall instead of a concrete structure, framing the move as a concession to Democrats.

Pence additionally confirmed that he would brief GOP leaders on Capitol Hill Tuesday before Trump's planned 9:00 p.m. ET national address on the shutdown from the Oval Office. The sit-down is expected to occur at approximately 5:30 p.m. ET.

"There is an ongoing humanitarian and a national-security crisis at our border," Pence said. "The president will address that in his remarks to the nation tomorrow night.”

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Meanwhile, Russell T. Vought, acting director of the White House Office of Management and Budget, assured reporters that "I can say unequivocally that tax refunds will go out as planned" even if the shutdown persists.

However, the Internal Revenue Service (IRS) is one of the agencies affected by the ongoing shutdown, and many IRS workers are among the nearly 800,000 federal workers currently furloughed.

For her part, Secretary of Homeland Security Kirstjen Nielsen, who reportedly clashed with House Speaker Nancy Pelosi, D-Calif., while trying to brief congressional leaders on border-security intelligence last week ("I reject your facts," Pelosi retorted, according to The Wall Street Journal), doubled down Monday on the administration's claims that terrorists have entered the U.S. through the southern border.

“We absolutely have had cases of terrorists crossing the southern border," Nielsen said. "The number itself is sensitive and difficult to share because some of the figures are classified, due to ongoing investigations. We can also say there are literally thousands of known or suspected terrorists traveling throughout the hemisphere, and if we don’t have a secure border, it makes it that much more difficult to tell who is coming in.”

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“Also, 17,000 people with previous criminal convictions were stopped at the border last year," Nielsen added. "We have a duty to understand who is coming into our country."

The White House's claims on the terror threat at the border have emerged as a major point of contention in recent days. Speaking to "Fox News Sunday," White House Press Secretary Sarah Sanders remarked that "nearly 4,000 known or suspected terrorists come into our country illegally, and we know that our most vulnerable point of entry is at our southern border."

But in a moment widely shared on social media, anchor Chris Wallace interjected, noting most of them have been captured at airports -- not at the border -- and that the State Department -- unlike DHS -- has said there's been no credible evidence of a terrorist coming across the southern border.

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"Not always," Sanders said. "But certainly a large number [was apprehended at airports]. It's by air, it's by land, and it's by sea. It's all of the above. ... The most vulnerable point of entry that we have into this country is the southern border."

Wallace additionally challenged Sanders on the Trump administration's claims that many "special interest aliens" have been stopped at the border. So-called special interest aliens, Wallace pointedly reminded Sanders, have come from countries known to produce terrorists, but not necessarily were terrorists themselves.

Fox News' Matt Leach and Kevin Corke contributed to this report.