As reported in later editions of last night's Evening Times, the pair, aged 61 and 23, were seriously assaulted on the No. 61 as it left the East End.
There are fears the father has been scarred for life, and he has had facial surgery.
The victims got on the bus at Tollcross Road, near Braidfauld Street, heading to the city centre at about 11.45pm on Monday.
Soon after, a man and woman got on at Helenvale Street, near Tollcross Road, and an argument took place involving all four.
Police said the victims were seriously injured when they were attacked by the male.
All four passengers left the bus at Gallow-gate, near Bellgrove Street, across from the Wee Man's pub.
The male attacker headed off towards Millroad Gardens and the woman walked east into the Gallowgate.
The victims went to their home nearby and raised the alarm.
They were both taken to the city's Royal Infirmary for treatment. Police said the 61-year-old needed surgery and the 23-year-old was treated and released.
Detective Constable Jim Mitchell, of London Road CID said: "They were viciously attacked."
Anyone with information can call CID on 0141 532 4600, or Crimestoppers on 0800 555 111.