About 15 animals were found dead, cut-up and decapitated, close to an intersection in the northeast of the city on Sunday.
Animal-cruelty investigators are working the case, after a resident of El Paso alerted authorities to the finding.
Captain Ray Spears, of the Texas Parks and Wildlife Department (TPWD), told news channel KVIA: "It did not look like the portions for eating were removed from the goats. The bodies seem intact".
He told the station: "Satanic rituals, blood removal, possibly. Not a common occurrence to see that many goats dumped like that."
TPWD confirmed to The Independent on Wednesday that the animals were domestic goats, but said it was not involved in the investigation.
A spokesperson for El Paso police, Enrique Carrillo, told the El Paso Times on Tuesday that its Animal Cruelty Investigations Unit was assigned the investigation.
“Anything beyond that at this point is purely conjecture and not based on any investigative conclusions," Sgt Carrillo added.
According to Capt Spears, the area is "popular for dumping because there are not many people to see them, especially at night. Pull up, drag them off the back of a pickup, and get out of there."
The Independent has approached the EPPD for further comment.
Although goats are often associated with sacrificial rituals, including in the Old Testament, the Church of Satan on Tuesday ruled out any responsibility for the El Paso incident.
"This is not a Satanic ritual. Satanism is an atheistic religion which directly prohibits harming animals, as such there are no rituals which involve sacrificing anything,” the Church of Satan wrote on Twitter. “This claim is irresponsible and puts people in danger”.