The pope's nephew, Emanuel Horacio Bergoglio, who was behind the wheel at the time of the crash, was in serious condition, they said.
His children -- aged eight months and two-year-old -- were killed in the crash along with their mother, Cordoba police commissioner Carina Ferreyra told AFP.
Vatican spokesman Federico Lombardi said the pope had been informed about the accident and was "deeply pained."
"He asks all those who share in his pain to join him in prayer," Lombardi added.
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The circumstances surrounding the crash were under investigation.
Bergoglio, who was hospitalized with multiple injuries, is the 38-year-old son of Alberto, the pope's late brother.
Both he and his two-year-old son were pulled from the wreckage alive and taken to hospital some 30 kilometers (19 miles) away where the child died shortly after being admitted.
"The patient has a guarded prognosis," hospital director Liliana Abraham told Continental radio.
Sergio Arroyo, another hospital official, said Bergoglio was admitted around 1 am with multiple trauma, an exposed fracture of the right humerus and a liver injury, which was operated on.
The truck driver, a man in his 60s, was not hurt.
The accident happened some 550 kilometres northwest of the capital Buenos Aires near the city of James Craik.
Fire Chief Gustavo Ponce de Leon told the TN television channel that Bergoglio was conscious when he was rescued and that he was at the wheel while his wife and children were in the backseat.