A spokesman for the army's land regiments based in the Isere region told Reuters earlier soldiers had told police the driver first passed them shouting abuse in Arabic, before returning and trying to run them down.
A spokesman for the French army said that a driver had insulted two groups of soldiers jogging in the road near their barracks in Varces-Allières-et-Risset.
The driver is still on the run; while a second individual who was in the same vehicle is also wanted.
A French judicial police officer stands at the entrance to the barracks in Varces-Allieres-et-Risset on March 29, 2018, after a man tried to ram his auto into soldiers jogging nearby.
"The soldiers managed to get up onto the pavement without being hit", Brulon said.
Quoted by the newspaper, the public prosecutor in Grenoble, Jean-Yves Coquillat, said that "we must remain cautious regarding the motives of the attacker".
Police sealed off the area and began a search for the driver, while the army stepped up security around the barracks, Brulon said.
France remains on edge after a terrorist went on a killing spree the south west last Friday, leaving four dead.
The attack follows a hostage-taking in the southwest of France on 23 March, in which four people were killed.
President Emmanuel Macron on Wednesday led a national tribute to Beltrame, who swapped places with a woman Lakdim had been holding hostage.
Lakdim was shot dead by police.
Over 240 people have been killed in a series of jihadist attacks around France over the past three years, and the security forces have been targeted on several occasions.
In August 2017, a man rammed his vehicle into a group of soldiers on anti-terror patrol in the western Paris suburb of Levallois-Perret, injuring six people.