Four men flew from Istanbul through Paris to Mexico City in late August, where they were met by a Turkish-speaking man who stashed them in a safe house until their Sept. 3 attempt to cross into the U.S. over the border with Mexico.

Their capture by the Border Patrol in Texas set off a fierce debate over the men’s intentions, with some members of Congress saying they were terrorist fighters. Homeland Security officials, including Secretary Jeh Johnson, countered that they were part of the Kurdish resistance which, like the U.S., is fighting the Islamic State’s advance in Iraq.

But whether the men are linked to anti-U.S. jihadists or not, they admitted to being part of a U.S.-designated terrorist group, and their ability to get into the U.S. through the southern border — they paid $8,000 each to be smuggled into Texas — details the existence of a network capable of bringing terrorists across the border.

The four men’s story, as discerned from internal September and October documents reviewed by The Washington Times, also seems to contrast with what Mr. Johnson told Congress in September, when he assured lawmakers that the four men were not considered terrorist threats to the U.S., even as behind the scenes his department proposed the four be put on terrorist watch lists.

Homeland Security spokeswoman Marsha Catron said the individuals weren’t associated with the Islamic State, which is also known by the acronyms ISIL and ISIS.

“The suggestion that individuals who have ties to ISIL have been apprehended at the southwest border is categorically false, and not supported by any credible intelligence or the facts on the ground,” Ms. Catron said. “DHS continues to have no credible intelligence to suggest terrorist organizations are actively plotting to cross the southwest border.”

She did not reply to questions about the status of the four men or why her department proposed they be put on terrorist watch lists.

As of a month ago they were being held at the South Texas Detention Facility in Pearsall, Texas.

The men initially claimed to be members of the Revolutionary People’s Liberation Party/Front, known by the acronym DHKP/C. The group is a Marxist insurgency that claimed credit for a 2013 suicide bomb attack on the U.S. embassy in Ankara, Turkey’s capital, last year.

But U.S. counterterrorism officials said the men were more likely members of the PKK, or Kurdistan Workers’ Party, which has been battling for Kurdish rights within Turkey for decades, though recently PKK and Turkish leaders have tried to broker a political agreement.

Both the PKK and DHKP/C are listed by the State Department as terrorist groups.

Jessica Vaughan, policy studies director at the Center for Immigration Studies, said the fact that avowed members of terrorist groups got into the U.S. shows it’s possible to sneak across a porous border.

“This incident proves what enforcement experts have always known, and that is there are existing networks in Mexico and Central America that have been set up and cultivated by a variety of terrorist organizations to enable them to move people into the United States illegally,” Ms. Vaughan said.

It’s unclear what the men were trying to do. None of them admitted to being part of a plot against the U.S., and several told investigators they were hoping to seek asylum, saying they believed they were being targeted back home by police in Turkey.

Two of the men had tried to enter the U.S. before but had been denied visas when they couldn’t prove they intended to return to Turkey, the documents showed.

Jonathan Schanzer, vice president of research at the Foundation for Defense of Democracies, said he would be surprised if the men were part of a plot against the U.S.

“We have the PKK listed as a terrorist organization, but we’re certainly not at war with the PKK the way we are with al Qaeda,” Mr. Schanzer said. “The possibility that the PKK could be trying to fundraise here in the United States or to be involved in political activity is certainly possible. But this seems like a very difficult way of trying to achieve that.”

He said the same was true of the DHKP/C: “It’s a Marxist-Leninist group. We’re not at the peak of the Cold War here anymore. This is a Turkish-oriented organization. It just does not seem to add up, from my perspective.”

Turkey is increasingly under scrutiny as a potential gateway for foreign fighters who have flocked to Syria and Iraq to fight the Assad regime or to join up with the Islamic State.

In an op-ed for Time on Wednesday, House Homeland Security Chairman Michael T. McCaul singled out Turkey, saying it doesn’t do a good enough job scanning passengers to figure out whether they are on terrorist watch lists.

“Equally worrisome is that terrorists might use refugee groups as a Trojan Horse to get into the West,” Mr. McCaul wrote. “Hundreds of thousands of Syrian refugees have poured into Turkey this year, and many of them have left for other European countries by boat. It is unclear whether extremists are hiding in these groups, as few are comprehensively screened on the way out.”

Mr. Schanzer at the Foundation for Defense of Democracies said Istanbul’s international airport was the hub for many foreign fighters arriving in the region, who then traveled to Syria or Iraq. He said the U.S. and other nations now need to begin wondering about the reverse path from Syria and Iraq back through Istanbul to their home countries.

“They’re going into Istanbul and then fighting, crossing the eastern front, and eventually they’re going to come back home,” he said.

Some members of Congress have called for the U.S. to revoke passports of American citizens who traveled to fight with the Islamic State, or to suspend its visa waiver program with countries that have large numbers of their citizens who went to join the fight.

President Obama has not followed those suggestions, though the Homeland Security Department this month did begin to demand more information from passengers traveling from visa waiver countries.

In early October, Rep. Duncan Hunter, California Republican, said he’d learned that foreign fighters tied to the Islamic State had been caught trying to cross the U.S.-Mexico border. He later clarified that they were “terrorist Turks.”

Rep. Jason Chaffetz also raised the possibility at a September hearing, asking Mr. Johnson whether he was aware of four men with terrorist ties were caught trying to cross the border into Texas earlier that month.

Mr. Johnson at the time said he didn’t know how much credence to give to those claims. But in early October, just after Mr. Hunter’s statement, Mr. Johnson delivered a reply in a speech to the Center for Strategic and International Studies on Oct. 9, saying that the four men caught were essentially the good guys.

“These four individuals were arrested, their supposed link to terrorism was thoroughly investigated and checked and, in the end, amounted to a claim by the individuals themselves that they were members of the Kurdish Worker’s Party — an organization that is actually fighting against ISIL and defended Kurdish territory in Iraq,” he said. “Nevertheless, these individuals have been arrested for unlawful entry, they are detained, and they will be deported.”

At the time he spoke, according to documents The Times reviewed, several of the men were still claiming to be DHKP/C rather than PKK.

The documents show they arrived in Mexico City on Aug. 23 and were met by a man who spoke Turkish and who took them to a stash house. They stayed there until Sept. 3, when they were driven from Mexico City to Reynosa, which sits on the Mexican side of the Rio Grande. The men crossed into Texas from there.

They were kept in a stash house in Mission, Texas, until they said they escaped Sept. 10. Two of them turned themselves into local police, who turned them over to the Border Patrol, who then captured the other two men.

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