The 51-year-old was attacked in Mount Florida in the early hours.
She needed treatment for facial injuries at the Victoria Infirmary following the attack.
Police said that around 12.30am the woman was walking outside the football stadium towards Somerville Drive when the man approached her.
It is believed he was at a Christmas party at Hampden and came out of the main entrance just before midnight.
CCTV suggests he spoke to another man who may have given him directions.
Police want to speak to the man who spoke to the suspected attacker and are also appealing for witnesses who were in the area of the stadium, Somerville Drive and Cathcart Road from 12.30am.
The woman needed treatment for facial injuries at the Victoria Infirmary following the attack, which happened on Sunday, December 8.
The suspect is white, about 30, 5ft 10ins to 6ft, and of medium build with short dark hair.
He was smartly dressed and was wearing a dark three-quarter length coat with distinctive buttons - possibly cream or white, a dark suit with a handkerchief in the breast pocket and a light shirt.
Police said they had increased their patrols in the area for public reassurance.
Detective Inspector Kate Jamieson, of the National Rape Task Force, said: "Extensive inquiries are continuing to identify the suspect, and we will trace this man.
"But I would appreciate any information from others who were at the Christmas party night who may know the man responsible, or recognise the description.
"I would also like to urgently trace the man who provided directions to the suspect.
"The woman is extremely traumatised by what has happened and is receiving the support she needs.
"I would urge anyone who recognises the description of the suspect, or has any information at all that will assist this investigation, to contact us as a matter of urgency.
"Even if you are unsure if the information you hold is relevant, please contact us so we can make that decision."
Anyone with information is asked to call police on 101, or Crimestoppers on 0800 555111.