Republican Sens. Chuck Grassley and Mike Lee are urging Department of Homeland Security head Kirstjen Nielsen to execute a "safe third country agreement" with the Mexican government, which would require members of the growing migrant caravan to stop and seek asylum there.

Such an arrangement requiring asylum seekers to make their claims in their first country of arrival, the senators noted, would also take some pressure off the overworked asylum system in the U.S., which has been increasingly inundated with claims -- the "vast majority" of which Trump administration officials have called fraudulent or legally dubious -- that judges must adjudicate.

A "safe third country" agreement with the incoming Mexican administration, which has vowed to be more sympathetic toward migrants, would not unprecedented. The senators, who sent a letter to Nielsen on Tuesday, cited the 2002 Canada-U.S. Safe Third Country Agreement, which requires that asylum applicants, absent extraordinary circumstances, request refugee protection in the first safe country in which they arrive.

SESSIONS ANNOUNCES PLAN TO HANDLE FRAUDULENT ASYLUM CLAIMS

"A safe third country agreement with Mexico would also address national security concerns inherent in a large group of unidentified individuals moving unabated toward the United States," the senators wrote.

They added that Jeh Johnson, President Barack Obama's Homeland Security secretary, wrote in a memo that "we must continually evaluate our border and port of entry security posture to ensure our resources are appropriately aligned to address trends in the migration" of "Special Interest Aliens" -- meaning those who are considered national security risks to the U.S. and attempt to gain illegal entry.

The called-for arrangement is not new. A Homeland Security official, referring to the request, said, “This is an avenue the Department has been [pursuing] with Mexico for more than a year.”

Homeland Security officials said Tuesday that among the thousands of migrants heading for the United States border are some “gang members” and people with “significant criminal histories.”

Department spokesperson Tyler Houlton said on Twitter that the caravan includes citizens not only of Central American countries, but also of the Middle East, Africa, South Asia and “elsewhere.”

“.@DHSgov can confirm that there are individuals within the caravan who are gang members or have significant criminal histories,” another tweet read. He did not offer more specifics.

And President Trump has in recent days flagged what he has characterized as national security risks posed by the caravan, whose membership has swelled to more than 7,000 people as it surges north through hot and humid Mexico toward the U.S.

The president told USA Today in an interview Monday that "people from the Middle East” are among the thousands of migrants in the caravan, echoing remarks he made on Twitter Monday morning, when he complained there were MS-13 gang "criminals and unknown Middle Easterners mixed in."

The president did not elaborate.

His comments drew criticism from several commentators, and a chryon on CNN on Tuesday flatly stated that Trump was "spreading lies and stoking fears" in a "fact-free campaign of fear."

Lee and Grassley noted in their letter that a 2017 report in Politico uncovered a scheme by Hezbollah, the Islamist militant group, to traffic "narcotics, weapons and persons" from Venezuela and Latin America into the U.S., and they pointed to Guatemala's recent disclosure that it had caught 100 ISIS terrorists there.

"For over a decade now, United States sovereignty has been tested and too often disregarded," the senators wrote. "Under President Trump's Administration ... the United States is finally in a position to secure our border and reinstitute law and order."

The letter comes as both Republicans and Democrats scramble to address the caravan with the Nov. 6 midterms fast approaching.

Trump has likened the caravan to an illegal immigrant invasion, and has pledged to cut aid to Central American countries and summon as many troops as necessary to the Mexican border.

In the Oval Office on Tuesday, Trump seemed to be accusing Democrats of possibly funding the caravan.

Sen. Ben Cardin, D-Md., said on Tuesday that the U.S. should "be out there and try[ing] to help" the caravan -- although Cardin did not say the migrants should be guaranteed entry.

Many migrants have said they've been escaping widespread violence, poverty and corruption in Honduras.

Still, the White House has framed the caravan as a potential importer of violence to the U.S., carrying on a theme it has emphasized in the past two years. In 2017, federal authorities arrested 796 MS-13 gang members nationwide as part of a deliberate crackdown on illegal immigrant gangs, and numerous cities have joined the Trump administration's lawsuit targeting sanctuary policies that shelter illegal immigrants.

Fox News' Paulina Dedaj contributed to this report.