That claim echoes earlier ones from Mr. Trump: baseless warnings that “unknown Middle Easterners” had infiltrated a migrant caravan and that terrorists were pouring across the southern border.

Nowhere in the White House’s 25-page counterterrorism policy, released in October, was the threat of terrorists infiltrating the nation’s southwest border raised. And the State Department, in a September report, said there was “no credible evidence” that terrorist groups had sent operatives to enter the United States through Mexico.

The Department of Homeland Security did not respond to requests for evidence of Mr. Trump’s claims.

The Washington Examiner report, like most of its predecessors over the past decade, did not include any photographic evidence of the prayer rugs in question and largely relied on hearsay.

“Along the Mexican border there have been stories of suspicious items picked up by local residents, including Muslim prayer rugs and notebooks written in both Arabic and Spanish,” former Senator Kay Bailey Hutchison, Republican of Texas, warned in a March 2005 speech.

Later that year, former Representative Tom Tancredo, Republican of Colorado, told CBS that, during a visit to the border in Arizona, “we’ve found copies of the Quran, we have found prayer rugs, we have found a lot of stuff written in Arabic, so it’s not just people from Mexico coming across that border.”

In 2014, David Dewhurst, then the lieutenant governor of Texas, invoked prayer rugs found “on the Texas side of the border in the brush.” PolitiFact Texas rated his claim “Pants on Fire,” and noted that it could find only one photo of a purported prayer rug, presented by the conservative news outlet Breitbart.

That photo, eight scholars and religious figures said, looked nothing like a prayer rug, and it appears to have been removed from the current version of the Breitbart article. (Perhaps, suggested Gawker, the photo was actually of an Adidas soccer jersey.)