The 19-year-old was walking in West George Street with her sister, 20, and friend, 22, at about 3am.
They were approaching the junction with George Square when a man came up to them and tried to grab the bag, but was unsuccessful. He ran off towards Buchanan Street.
The woman was not hurt but police said she had been "very shaken" by the incident.
Officers are appealing for any witnesses to contact them and are studying CCTV footage as part of their inquiries.
Constable David Adam, of the robbery unit in Pollok, said it was a "frightening" experience for the woman.
"I would appeal to anyone who was in or around West George Street near George Square at that time on Saturday morning, particularly a man who was using a cash machine in West George Street just after the incident.
"We have information the would-be thief was near this man while he was using the machine. He may have information that could assist us.
"At that time people would have been leaving clubs and pubs, so we would ask anyone who may have seen a man acting suspiciously and fitting the description of the suspect to call us."
Police said the would-be thief was Asian, about 5ft 7in-5ft 9in and of slim build, with spiky hair. He was wearing a dark long sleeved top and light jeans or chinos.
Anyone with information is asked to call 101, the non-emergency number.