Mexico considers the El Paso massacre a "terrorist" act and is looking at legal action to extradite the gunman, Foreign Minister Marcelo Ebrard said.
Seven of the 20 people killed in Saturday's shooting at a Walmart store were Mexican.
Mexico's attorney general was considering possible legal action and extradition of the suspected shooter, who surrendered to police a block from the El Paso store, Mr Ebrard told reporters.
"For Mexico, this individual is a terrorist," he said.
Separately, President Andres Manuel Lopez Obrador "firmly" demanded that those responsible be punished and criticized the indiscriminate use of weapons in the United States.
"The authorities must also assume responsibility in the event that excesses are allowed, such as the indiscriminate use of weapons," he said at a public event in the central state of Michoacan.
Mr Ebrard said on Twitter he would travel to El Paso on Monday to meet with those affected by the shooting "and give them the full support of the Government of Mexico."
He added that the foreign ministry would also file a suit against those responsible for selling the assault weapon to the gunman and would inquire whether the authorities were "aware of the potential of this individual."
Mexico will also deliver a diplomatic note to Washington today asking it "to set a clear and forceful position against hate crimes."
"Nothing is solved by violence and nothing is solved by what is called xenophobia, hatred of foreigners, hatred of migrants," said Lopez Obrador.
Ebrard expressed "his deepest rejection and condemnation" of the "barbaric act" committed in El Paso.
The gunman, identified in media as a 21-year-old white man named Patrick Crusius, is believed to have posted online a manifesto denouncing a "Hispanic invasion" of Texas.
El Paso, on the border with Mexico, is majority Latino.