Agent Rogelio Martinez's body was found on the side of a busy roadway near the Mexican border outside El Paso, Texas on November 18.
Art Del Cueto, national spokesman for the union, said Wednesday the FBI's comments were preliminary.
"The dispatcher made a notation in the Border Patrol log "[He] thinks they (both agents) ran into a culvert", according to the report. It said it had conducted 650 interviews and 26 searches in Texas and New Mexico.
The office of Gov. Greg Abbott (R-Texas) similarly called Martinez's death a "murder case" and offered a reward of up to $20,000 for information "leading to the arrest and conviction of the person or persons responsible for the murder of a U.S. Border Patrol agent and the serious injury of another".
To date none of the more than 650 interviews completed, locations searched, or evidence collected and analyzed have produced evidence that would support the existence of a scuffle, altercation, or attack on November 18, 2017.
Authorities were also alerted by Garland's wife, who called the Van Horn Border Patrol station saying she'd received a cell phone call in which her husband seemed confused and disoriented, according to an agent who spoke with her. Martinez was found November 19 with fellow agent Stephen Garland, who had also been seriously injured - but survived the incident. "This is a stark reminder of the ongoing threat that an unsecure border poses to the safety of our community and those charged with defending them".
A medical examiner in Texas said Martinez died from blunt injuries to his head, but the manner of his death is "undetermined," according to an autopsy report released Tuesday. The dispatcher noted that the second agent said something to the effect of "We ran into a culvert", "I ran into a culvert", or "I think I ran into a culvert" during the dispatch call.
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Martinez's fiancée, Angie Ochoa, said she believes the autopsy report is cover-up. One of them, Rogelio Martinez, died of his wounds.
Brandon Judd, president of the National Border Patrol Council, the agents' union, concluded the agents were brutally beaten with rocks.
"The absence of evidence is a key factor in this case - not due to lack of effort or determination, but because evidence which would indicate the presence of other persons or the commission of a criminal act is not present", the memo reads.
The FBI says the investigation remains open and inconclusive, but its statement counters hasty conclusions by some conservative politicians and law enforcement figures who assumed unauthorized immigrants committed a brutal murder in the lawless borderlands. They have found no evidence to support that. A forensic analysis later determined they had nothing to do with Martinez's death or the injuries to Garland.
The FBI has been considering several possibilities, including an accident, an attack or an altercation between the two agents, a Department of Justice official with knowledge of the investigation said.
The FBI will continue to investigate the incident involving Agent Martinez and his partner, and will pursue any new and relevant tips and leads. A reward of up to $50,000 is offered by the Federal Bureau of Investigation and an additional $20,000 is being offered by the State of Texas for information leading to a resolution in the case.