Up to eight Mexican tourists were said to be among the dead and Egypt’s acting prime minister Ibrahim Mehleb later offered his “deepest condolences”.
A joint military-police force was pursuing “terrorist elements” in the area and fired on four cars that turned out to be carrying tourists.
Mexican President Enrique Pena Nieto immediately condemned the deaths and demanded a full investigation. Egyptian authorities insist the convoy had entered a restricted area of the desert without permission and without informing authorities.
Mehleb, whose government resigned last week but who remains in his post until a successor can be chosen, visited the wounded in Dar el-Fouad Hospital in Cairo.
“We are standing next to the Mexican people in their tragedy,” he said. “The circumstances surrounding the incident are part of Egypt’s fight against terrorism. The Mexican side is receiving the best care and we will take care of the Egyptian victims.
“This is a painful incident and I give my deepest condolences to the Egyptian people and our guest the Mexicans, and I have spoken to the Mexican ambassador and relayed my condolences.”
It was reported the safari was organised by a local hotel and the tourists were preparing to camp out in the vast desert when they came under fire.
Rasha Azazi, a spokesperson for the Egyptian ministry of tourism, insisted the tour company involved “did not have permits and did not inform authorities”, adding that any trips to the Farafra area are required to be cleared by officials.
Mexican foreign secretary Claudia Ruiz Massieu said the wounded had told Mexico’s ambassador to Cairo that they were subject to an aerial attack from aircraft.
Mexico has sent a diplomatic note expressing “profound dismay” and demanding an investigation.
US Embassy spokesman Brian Shott, meanwhile, said officials were looking into whether an American citizen was involved in the incident.
Egypt has been battling an Islamic insurgency in northern Sinai for years, but attacks – mainly targeting the army and police – escalated and spread to the rest of the country after the July 2013 military ousting of Islamist president Mohammed Morsi.
Egypt’s western desert is popular among safari enthusiasts, but has not been known as a hotbed of militant or insurgent activity.
Police and military there have primarily been concerned with combatting smuggling along Egypt’s border with Libya.
On Sunday, the Islamic State group (IS) claimed it had “resisted a military operation” in the desert. A group claiming to be affiliated with IS also said that it was present in Farafra.