The incident happened at a property in Kelhead Path, Cardonald, at around 4:30am on 12 May.
Police say the man involved was between 30 and 35 years of age, 6ft tall, of a slim build and clean shaven. He was wearing all dark clothing and a dark-coloured skip cap.
Police said extra officers will be in the area at the weekend.
They also appealed for anyone who heard anything or saw anyone matching the description to get in touch.
Detective Inspector David Reilly, of the divisional rape investigation unit, said: “This was a horrendous ordeal for this woman to have to experience and it is critical we trace this man as soon as possible.
“I would like to reassure the community that our officers are doing everything they can to find the person responsible, including checking footage from extensive CCTV cameras covering the area.”
DI Reilly added: “However, we need help from members of the public who may have vital information to assist with the investigation.
“Did you hear anything, or see anyone matching the description in the early hours of Saturday morning in the area? Perhaps you were woken up, or still awake at that time.
“I would appeal to these people to please contact us and pass on any small piece of information.
“You may think it’s not important, but it could turn out to be that vital piece of information which leads us to the suspect.
“We would advise anyone who is at all suspicious of people who call at your home, and has any concerns for their own safety to call police via 101 or in an emergency, 999.”
He added: “Extra police officers will be in the area over the weekend and any concerns can also be reported to them directly.”
Anyone with information about the incident or the man involved should contact Police Scotland on 101.