A U.S. Border Patrol agent scans the U.S.-Mexico border while on a bridge over the Rio Grande on March 13. (John Moore/Getty Images)

Along the shoulder of Interstate 10, where drug runners sometimes hide in drainage culverts, U.S. Border Patrol agent Rogelio Martinez came to a stop late Saturday and got out of his vehicle.

Martinez scrambled down an embankment through ocotillo and creosote bush, splitting the darkness with his flashlight. Then, according to Border Patrol union officials, attackers struck him from behind.

Martinez, 36, was found dying moments later, apparently bludgeoned with rocks, the union officials say. Nearby was another agent who arrived with Martinez or soon after, badly beaten but alive, the officials say.

The attack described by officials would be the first killing of an on-duty Border Patrol agent since the 2010 murder of Brian Terry in Arizona. That event was a catalyzing moment in the push for tougher border enforcement, and President Trump has cited Martinez's death to boost his case for a wall along the boundary with Mexico.

"We will, and must, build the Wall!" he tweeted, saying Martinez had been "killed."

But the FBI, which is leading the investigation of Martinez's death, released a statement late Monday that did not confirm the agents were attacked. The two men were found at 11:20 p.m. Saturday in a culvert area about 12 miles east of Van Horn, the statement said, with traumatic head injuries "along with other miscellaneous physical injuries such as broken bones."

Both agents were airlifted to El Paso, where Martinez died from his injuries early Sunday, according to the FBI statement. "The results of an autopsy are pending," the statement said, adding that the second agent remains in intensive care but in stable condition.

The local sheriff expressed skepticism about how the incident was described by union officials and said the agents may have fallen.

"The evidence is not obvious as to what happened out there," Culberson County Sheriff Oscar Carrillo said in an interview at his office in Van Horn, population 2,500.

Carrillo said that Saturday was a moonless night and that the agents were walking along difficult terrain with deep culverts.

"The injuries to [Martinez], after talking to his doctors, were consistent with a fall. Very consistent with a fall," Carrillo said. "Why is the other guy hurt? That's the unknown. What happened to him?"

Authorities have not identified the injured agent, who remains in the hospital in serious condition, officials said.

A spokesman for the Department of Homeland Security, which oversees U.S. Customs and Border Protection, said it was waiting for the results of the FBI probe. The Border Patrol union does not speak for the agency, he said.

Brandon Judd, president of the National Border Patrol Council, the agency's union, said in an interview Monday that Martinez died of devastating blunt force trauma to the head.

"I have been told by several agents that it was a grisly scene and that his injuries were very extensive," Judd said Monday in a phone interview. "We believe he was struck in the head with rocks, or multiple rocks."

According to Judd, the area where Martinez was allegedly attacked is about 50 miles north of the border, along a smuggling corridor often used by drug couriers carrying backpacks full of marijuana.

Martinez, an El Paso native, had been a Border Patrol agent since 2013. He called on his radio to report suspicious activity, Judd said, then began tracking what agents refer to as "signs," typically footprints or other evidence of a recent disturbance to the soil or vegetation.

At the same time or soon after, a second Border Patrol agent arrived in a separate vehicle to provide Martinez with backup, Judd said. That's standard procedure, he added.

Judd said Martinez would have been an easy target for attackers, as he likely would have been walking with his flashlight out and head down to follow the footprints.

"There was no appearance that either agent went for their weapon," Judd said, adding that the alleged attackers did not attempt to take the agents' guns.

Martinez never regained consciousness, according to Judd.

The second agent drifted in and out of consciousness, and Judd said his memory of the attack remains foggy. "So there is a possibility they were together when this happened, but we're getting a lot of different information from different agents."

Texas Gov. Greg Abbott (R) announced a $20,000 reward for information leading to an arrest and conviction for what he called the "murder" of a Border Patrol agent.

Sen. Ted Cruz (R-Tex.) also linked the incident to security on the border with Mexico.

"This is a stark reminder of the ongoing threat that an unsecure border poses to the safety of our communities and those charged with defending them," Cruz said on Twitter.

The area is a lonesome stretch of highway about 100 miles east of El Paso. It is part of U.S. Customs and Border Protection's vast Big Bend Sector, which includes 135,000 square miles in Texas, including 510 miles of river border. The sector's Van Horn Station, near where Martinez was injured, covers 15 miles of the Mexico border.

Border Patrol statistics show that the Big Bend Sector accounted for less than 2 percent of apprehensions during the government's 2016 fiscal year, indicating that the desolate area is not a major corridor for illegal migration. But among the agency's 14 border sectors, Big Bend was fifth-highest for marijuana seizures.

Terry's 2010 murder remains a rallying cry for proponents of a wall, and just hours before Martinez's death, former White House strategist Stephen K. Bannon accepted a courage in journalism award from a foundation started by the slain agent's family.

"Brian Terry will live in history as a historical figure," Bannon, who heads the right-wing Breitbart News site, told about 300 guests, "and the reason was, he brought to the attention of the American people, he put a human face on it, he put a hero's face on it, of what is happening on the southern border." Trump would not have been president were it not for Terry's death, Bannon said.

Bever and Miroff reported from Washington. Derek Hawkins and Devlin Barrett in Washington contributed to this report.