As of Sunday morning, Martinez's partner, whose name has not been released due to privacy concerns, is in serious condition. Ted Cruz (R-TX) released a statement calling it a "stark reminder of the ongoing threat that an unsecured border poses to the safety of our communities and those charged with defending them".
Jeanette Harper, a spokeswoman for the FBI's El Paso office, told the San Antonio Express-News the agents were not attacked with guns.
"Our thoughts and prayers are with Agent Martinez and his family, and with the agent who was injured", CBP said in the statement. He says the wall he has promised to build along the border between the USA and Mexico is on the agenda. Trump promises to find the perpetrators and seek justice, as well as to build a wall along the Mexican border.
"We're going to straighten it out", he said.
Authorities are scrambling to find whoever killed a Border Patrol agent and hope a $20,000 reward will help lead them to the assailant.
"Agent Martinez's partner reported that they were both injured and in need of assistance".
"We are grateful for the courage and sacrifice of our border agents who have dedicated their lives to keeping us safe", Cruz said.More news: Fort Hood leads with most sexual assaults reported in US Army posts
Abbott also described the incident as an attack in a tweet Sunday. "The men and women of our Border Patrol have an incredibly risky and demanding job and they are our most important resource to securing our border".
Customs officials would not say what type of activity the agents were responding to.
Customs and Border Patrol spokesman Carlos Diaz declined to describe the incident in detail, saying the FBI is investigating.
U.S. Customs and Border Protection issued a statement Sunday that was thin on details about what happened, saying the two agents "were responding to activity" while on patrol near Interstate 10 in the area of Van Horn, which is about 30 miles from the border with Mexico and about 110 miles southeast of El Paso.
Authorities were searching Texas' Big Bend area for suspects and witnesses Monday after US Customs and Border Protection agent Rogelio Martinez was allegedly bashed to death with a rock while on patrol.
Martinez had been a border patrol agent since 2013, officials said.
Border patrol records show that the Big Bend region, where the apparent attack took place, only accounts for 1% of apprehended migrants. The region's mountains make it a hard area for people to cross illegally into the US from Mexico.