The 31-year-old and the six-year-old were outside Glencairn Social Club on Glasgow Road in Rutherglen at the time of the incident, which occurred at around 12:15am on Sunday.
Police Scotland said the Volkswagen Golf private hire taxi failed to stop. The 60-year-old male driver was traced a short time later.
The man’s injuries are described as life threatening, while the boy is said to be in a stable condition. They are being treated at the Queen Elizabeth University Hospital.
It is understood there was a function on at the Glencairn on Saturday night, but a member of staff reached at the club declined to comment.
Sergeant Craig McDonald said: “Inquiries are at an early stage to establish the exact circumstances of this incident and we want to speak to anyone who may have witnessed what happened.”
The incident comes just a week after two other teenagers were struck by a car that then drove off. Police are still appealing for witnesses to the case, which they are treating as attempted murder.
Around 2:15pm on Sunday, 24 January, two boys aged 14 and 15 were walking along Fernhill Road when a light green Vauxhall Corsa mounted the pavement and struck the 14-year-old. The vehicle then made off towards the Castlemilk area. The two teenagers made their way home and police were called. The younger boy was taken by ambulance to the Queen Elizabeth University Hospital, where he remains in a stable condition. The 15-year-old was not hurt.
Police recovered the Corsa a short time after the incident and inquiries are continuing to trace the who was in the car.
Anyone with information on either incident is asked to call police on 101 or Crimestoppers on 0800 555 111.
Rutherglen Councillor Robert Brown said: “It’s a dangerous problem in a number of residential streets where people drive significantly too fast.”