"We owe a great deal of gratitude to the fearless men and women of the United States Border Patrol who serve every day to protect our homeland", said Governor Abbott.
Border Patrol Rogelio Martinez died under mysterious circumstances while on duty on November 20, 2017. His partner, who has not been identified, remains in the hospital in serious condition.
Agent Rogelio Martinez and his partner were responding to activity while on patrol near Interstate 10 in the Van Horn Station area when Agent Martinez's partner reported that they were both injured and in need of assistance.
Authorities believe Martinez may have fallen into a culvert and said Martinez's partner actually radioed for help after the incident but has no memory of what occurred.
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An FBI spokeswoman told the San Antonio Express-News on Sunday that the agents were "not fired upon", but she didn't elaborate.
Border Protection spokesman Carlos Diaz confirmed to The Associated Press in an email that the FBI was in charge of the investigation.
Martinez had been a border agent since August 2013 and was from El Paso.
Authorities did not provide any details Sunday on what caused the agent's injuries or what led to them.
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Trump took to Twitter on Sunday to insist that Martinez's death underscores the need for a wall along the border between the USA and Mexico.
The FBI in El Paso announced Tuesday a $25,000 reward for any information that leads to the arrest and conviction of anyone possibly involved in the death of Border Patrol Agent Rogelio Martinez.
Earlier, the Republican tweeted that 'resources must be increased to prevent these attacks in the future'. We are grateful for the courage and sacrifice of our border agents who have dedicated their lives to keeping us safe. The president offered his condolences to Martinez's family.
'We will, and must, build the Wall!' he tweeted on Sunday. He also said Martinez's partner was "brutally beaten and badly, badly hurt" but that it "looks like he'll make it". The second injured Border Patrol agent is now in a local El Paso hospital's intensive care unit (ICU) in stable condition. The region's mountains and the Rio Grande make it a hard area for people to cross illegally into the US from Mexico.
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Martinez is the second agent to have died this year. Some were attacked while working along the border and others were killed in traffic accidents.