Eyewitnesses have described a Jeep Cherokee being driven at speed towards the building with flames coming out from underneath.
They have also described seeing two Asian men, one of whom was on fire, who had been in the car.
Strathclyde Police said two people had been arrested and detained in connection with the incident.
The airport has been evacuated and all flights suspended following the incident at 1515 BST.
A Whitehall spokesman said the incident was not being treated as a national security threat.
One eyewitness said: "I heard the sound of a car's wheels spinning and smoke coming out.
"I saw a Jeep Cherokee apparently as if it was trying to get right through the doors into the terminal building.
"There were flames coming out from underneath then some men appeared from in amongst the flames.
"The police ran over and the people started fighting with the police. I then heard what sounded like an explosion."
Eye-witness Richard Gray told BBC News 24: "A green Jeep was in the middle of the doorway burning.
"There was an Asian guy who was pulled out of the car by two police officers, who he was trying to fight off. They've got him on the ground.
"The car didn't actually explode. There were a few pops and bangs which presumably was the petrol."
Dr Rak Nandwani was at the airport building to pick up relatives when he saw a plume of black smoke.
He said: "The whole place has come to a standstill, the terminal building has been evacuated. I have spoken to my relatives and they have been moved to outside the building.
"They were told they could not pick their baggage up from the baggage collection area.
"There must be about 50 police cars at the airport. Me and my son, along with everyone else, have been moved to the car rental area."
Taxi driver Ian Crosby said: "This was no accident. This was a deliberate attack on Glasgow Airport."
The reaction of members of the public was not to help the men in the car, but to restrain them, he told the BBC.
Mr Crosby, who said he served in Northern Ireland, told how he shepherded people away from the scene in case of a secondary explosion or a nail-bomb.
Two men, one of whom was reported to be badly burned, were seen being led away in handcuffs.