The pair, aged 83 and 60, were in Glasgow Green when the incident took place at around 2.30pm on Friday.
A man approached them when they were outside the People’s Palace in Monteith Row and carried out a “violent and unprovoked” assault before running off towards London Road, police said.
The women went to Glasgow Royal Infirmary where they were treated for facial injuries.
The man is described as white, around 20 years old, 5ft 8in and of medium build. He was wearing dark clothing and carrying a dark rucksack.
Detective Constable Paul Fraser said: “This was a completely unprovoked and cowardly attack on an elderly woman and her daughter who had come to Glasgow to enjoy a day out.
“Both victims have been left very distressed by the ordeal and have returned home to Fife.
“Glasgow Green would been busy around the time of the incident and I would appeal to anyone who saw what happened to get in touch.
“In particular we would like to speak to the driver of a coach who witnessed the attack and went to their assistance. We would urge him to come forward as soon as possible.”