The incident happened in Beith Street, Partick, between 1am and 3:20am on Sunday, August 3.
A 29 year-old man has been arrested in connection with the incident and appeared in court the following day.
Meanwhile, officers have appealed to the occupants of a car near the scene at the time to come forward and assist with their enquiry.
Detective Inspector Liz McLean said: "From our investigations, we have established that there was a small dark coloured hatchback car in the area at the time with a man and woman on board - described as in their 20s - who may have witnessed the incident or who may have information that will help us with our enquiries.
"I would ask that the man and woman contact police at the Divisional Rape Investigation Unit at London Road on the non emergency number 101 or Crimestoppers on 0800 555 111 where details can be given in confidence."