The incident occurred on November 27 in neighboring Ingushetia.
A security officer was killed in the accident, but Alkhanov was unhurt.
Alkhanov's spokesman called it a "normal traffic accident."
Last year, Chechnya's then-prime minister, Sergei Abramov, was seriously injured when a truck collided with his car outside Moscow.
Officials called it an accident, but suspicions persisted it was an attempt to force Abramov out of power to pave the way for Ramzan Kadyrov, who subsequently became prime minister.
Ramzan Kadyrov Turns 30
'PLEASED, HAPPY, AND GRATEFUL': RFE/RL's North Caucasus Service interviewed a number of Chechens in the republic, in Moscow and Amsterdam. An overwhelming majority had nothing but praise for their prime minister -- and hope that he might soon become president:
"All of us are pleased, happy, and grateful," one woman said. "And all of us know perfectly well who should be thanked for that -- of course, Ramzan Kadyrov, who followed the noble path of his father [assassinated Chechen President Akhmed-hadji Kadyrov]."
"We should have such a birthday every six months. Because of his birthday, they are building roads, renting out apartments, patching up the holes. Today the airport is opening up! It would be hard to think up a greater joy for Chechens," a man in Chechnya said. "We have spent so much money on taxis in order to get to the airports in Sleptsovsk, Nazran, Nalchik. What has Ramzan done for us? We don't even know it all. Everything that God has given him, he has dedicated to the prosperity of the Chechen Republic. God help him. Let him have a birthday every six months. That's what I say."
"Why wouldn't people want him to become president? He's done a lot of work and provided a lot of help," another man said. "It's good that poor people have someone to help them. I think that poor people would want to see him as president."
Chechens further afield, however, had few warm words about Kadyrov. This woman spoke to RFE/RL from Amsterdam.
"I don't think he is an exceptional person," she said. "And I don't think that people are praising him because they have forgotten about their killed and tortured. I think that all this is being done to put an end to things, so that they would stop the killing. They think he can do this. They think that here he is tsar, judge, and God. So they are afraid. They are afraid to criticize him. The price of the airport that is being opened and the other buildings has been the blood of many people. What is built on blood cannot stand. History tells us how quickly people like him are forgotten -- so-called dictators after they are either chased out of power or killed. A lot of things will appear wrong, inadequate, disgusting when he is no more. I feel sorry for the people who live there. Most likely, they are praising him because they think they have no other choice."