The message from Kevin McAleenan, acting head of U.S. Customs and Border Protection, to his staff said an exhaustive probe into the death of agent Rogelio Martinez was still under way but had not yet turned up any signs he was the victim of a crime.
"We do not know all the answers at this time. However, according to the FBI, currently none of the completed interviews, locations searched, or collected and analyzed evidence have produced evidence that would support the existence of a scuffle, altercation or attack," McAleenan wrote in the memo.
Martinez and another agent who survived, Stephen Garland, were in sparsely populated Culberson County, about 130 miles (210 km) southeast of El Paso, on Nov. 19 when they suffered head injuries and broken bones.
Garland's injuries have left him unable to recall the incident.
Last week the FBI's El Paso Field Office said in a statement that after conducting 650 interviews it had found no evidence of an attack. An autopsy report released a day earlier showed that Martinez died of blunt-force trauma.
Shortly after Martinez's death, Trump tweeted: "Border Patrol Officer killed at Southern Border, another badly hurt. We will seek out and bring to justice those responsible. We will, and must, build the Wall!"
According to McAleenan's memo, both agents appeared to have fallen into a nine-foot-deep culvert near Interstate 10 at about 11:30 p.m. local time on a moonless night.
In late November, the U.S. Department of Justice offered a $25,000 reward for information related to a "potential assault on a federal officer" in the case.
Some 33 border agents have died on duty since the CBP was created in 2003, the libertarian think tank Cato Institute said in a report after Martinez' death. Of those, about half were from auto accidents and four were murdered, the institute said.
(Reporting by Bernie Woodall in Fort Lauderdale, Florida, and Dan Whitcomb in Los Angeles; Editing by Paul Simao, Alistair Bell and Daniel Wallis)