The man, who has not been named, has been accused of off-loading the passengers at the side of the road at 2am on a Monday morning after they objected to his demand that they stop speaking in Irish Gaelic.
Glasgow City Council enforcement officers are looking into a formal complaint against the Hampden Cabs driver.
A council spokesman said: "A complaint regarding an alleged racist incident has been received and will be investigated in the appropriate manner by the council's taxi enforcement team."
Kathleen McAleer, 21, a mental health nurse, was one of four passengers in the car in the early hours of Monday, December 16. She was travelling with a friend and two cousins, who are from County Donegal, from one house on the South Side to another.
The cousins, Joseph, 19, and Anthony Blair, 21, are native Irish Gaelic speakers .
Ms McAleer said: "My cousins were just talking to each other in Irish, which is their first language.
"The taxi driver turned around and said to them 'Stop speaking in that language'.
"We didn't really know how to take it. He said: 'When you are in Britain, it is English you speak.'
"I said to the driver 'Excuse me', shocked that somebody would say that to them. I said: 'That is out of order'.
"He then said: 'If they want to speak in that language they can get out of my taxi.' So we got out and said we wouldn't pay."
Ms McAleer reported the matter to both Hampden Cabs and the city council.
A spokesman for Hampden Cabs refused to comment.
Private hire and taxi drivers must prove they are a "fit and proper person" to hold a licence from the city's licensing committee. Investigations are carried out by the city council's enforcement team with any findings passed on to the committee for a hearing, should one be deemed necessary.