The White House said Tuesday it "couldn't possibly answer" whether terrorists had already crossed over the southern border amid the ongoing crisis there and increased security threats in the Middle East.

National Security Council spokesman John Kirby made the statement during the White House press briefing while being pressed by Fox News' Peter Doocy on what FBI Director Christopher Wray said earlier in the day was the terror threat to Americans reaching a '"whole other level."

"We are always concerned about the potential presence on U.S. soil of terrorists coming from overseas. That's something we're always worried about," Kirby said, initially dodging when asked whether the White House had considered the possibility that terrorists could already be in the country after crossing the border.

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John Kirby White House

National Security Council Coordinator for Strategic Communications John Kirby speaks during the daily briefing in the Brady Briefing Room of the White House in Washington, DC, on October 31, 2023. (ANDREW CABALLERO-REYNOLDS/AFP via Getty Images)

Doocy noted a Customs and Border Protection bulletin obtained by Fox News last week warning staff that the risk of foreign terrorist fighters engaged in the Israel-Hamas conflict could soon be attempting to enter the U.S. via the besieged southern border. 

He asked if the White House was feeling any "heartburn" around that considering there were 600,000 known got-aways that had crossed into the U.S., meaning they were seen or detected by Border Patrol agents but not apprehended.

"I can't speak to this intelligence report that was leaked to the media. I wouldn't do that. I can tell you that we are constantly monitoring as best we can, all ports of entry to the country for the potential arrival of anybody who might wish us harm," Kirby said before calling on Congress to pass a supplemental border funding package recently requested by the Biden administration.

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Migrant captured by Border Patrol

U.S. Border Patrol agents capture a migrant near the U.S. and Mexico border fence in Calexico, California, U.S., on Tuesday, Sept. 14, 2021. (Eric Thayer/Bloomberg via Getty Images)

"But if the general gist of your question is are we taking the potential threat seriously, of course we are," he said.

Doocy responded that the gist of his question was whether it was possible somebody wanting to commit a terror attack in the U.S. during a time of elevated threat had already crossed the southern border.

"I couldn't possibly answer that question, Peter. All I can do is tell you that we have remained vigilant to that potential threat," Kirby said.

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Fox News' Adam Shaw, Bill Melugin and Danielle Wallace contributed to this report.