In the August 3 shooting at a Walmart store, at least 20 people were killed and 26 were injured.
In a video posted on his official Twitter page, Ebrard called what happened in El Paso "unacceptable" and said "the first judicial actions" the government would take would be in accordance with global law.
"I know the death penalty is something very powerful, but in this occasion it's something that's necessary", El Paso District Attorney Jaime Esparza said during a press conference on Sunday morning, according to the Texas Tribune.
El Paso, which boasts an 83 per cent Hispanic or Latino population according to the USA census data, is a popular crossing point between the United States and Mexico.
In a statement, Ebrard said, "We consider this act an act of terrorism against the Mexican-American community and Mexican citizens living in the United States". Among the dead were six Mexican nationals, with several others injured.
Seven victims in the attack have been confirmed to be from Mexico, and President Andrés Manuel López Obrador reportedly wants to pursue legal as a way of protecting families and "demand that the United States protect the Mexican community in the United States". "For Mexico, this individual is a terrorist".
The Texas killings were followed just 13 hours later by another mass shooting in Dayton, Ohio, where a shooter in body armor and a mask killed nine people in less than a minute and wounded 27 others in the city's downtown historic district before he was shot dead by police. He also stated that the lawsuit will allow Mexico to access information of those threatening the Mexican population in the U.S.
For his target, the suspect chose one of the largest and safest cities on the US-Mexico border, a place central to the Trump administration's hardline stance on immigration and a city that state Rep. Cesar Blanco called "ground zero" of the administration's family separations policy.
The Minister said that the Mexican government will not use hate but will instead use legal measures to seek justice for those killed.
President Trump has not responded publicly to the threat of legal action. A 21-year-old male suspect was taken into custody in the city which sits along the U.S. -Mexico border. "Mexico is outraged. But we aren't proposing to meet hate with hate".