An Ingush official source told the RIA Novosti news agency that the attack on the convoy appeared to be a "well-planned ambush".
"It was fired upon from at least three different points with machine guns and grenade launchers," he said.
Concerns have grown in recent weeks about the stability of Ingushetia, one of Russia's most violent regions.
Yunus-Bek Yevkurov, the Ingushetian leader, was gravely wounded when a suicide car bombing hit his armoured car on June 22, and he remains in hospital.
After the attempt on Yevkurov's life, Dmitry Medvedev, the Russian president, ordered Chechen security forces to assist Ingushetia in batting fighters there.
Ramzan Kadyrov, Chechnya's Kremlin-backed leader, said he had given orders to be personally informed of all aspects of the investigation and characterise dthe ambush as a desperate act by anti-government fighters.
"All they can do to us today is crudely shoot us in the back from the bushes. And we are going to put an end to this," he said, RIA Novosti reported.
Kadyrov's harsh tactics have brought relative stability to Chechnya since he took power in 2007 after more than a decade of war, leading Moscow to declare an end to military operations in the republic in April.
Islamist fighters are battling pro-Kremlin authorities and Russian security forces in a low-level insurgency in the overwhelmingly Muslim regions of Chechnya, Dagestan and Ingushetia.