"On behalf of the quarter of a million front line officers and agents of DHS, my thoughts and prayers go out to the family and friends of Agent Martinez and to the agent who is in serious condition", acting Secretary of Homeland Security Elaine Duke said in the statement.
A Customs and Border Protection spokesperson declined to offer any further details about the incident.
The statement said Agent Rogelio Martinez, 36, and his partner, who was not named, were responding to activity in the Van Horn section near Interstate 10 when the incident happened. An FBI official also said that, counter to initial reports, neither agent was shot.
Authorities were scouring West Texas on Monday for those behind an apparent attack that killed one USA border agent and injured another. A spokesman with the El Paso Independent School District was unable to immediately confirm Martinez's enrollment at Irvin due to the district being closed for Thanksgiving week.
Both the identity of Martinez's partner and the nature of the pair's injuries have remained unspecified by the agency.
Illegal immigrants appeared to have "ambushed" two U.S. Border Patrol agents near the Texas border with Mexico and bashed their heads with blunt objects - possibly rocks - killing one agent and sending another to a hospital in serious condition Sunday, a National Border Patrol Council official told Fox News.
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Martinez, who was originally from El Paso, had worked as a border agent since 2013.
Judd said details remained foggy on what led to the ambush and the FBI was investigating the incident. Responding agents provided immediate medical care, and transported both agents to a local hospital.
On Monday, Gov. Greg Abbott's office announced a reward of up to $20,000 through Texas Crime Stoppers for information leading to the arrest and conviction of whoever killed Martinez and wounded his partner. His name wasn't released.
"Big Bend Sector was later told that Agent Martinez expired from his injuries".
The region's mountains and the Rio Grande make it a hard area for people to cross illegally into the United States from Mexico.