Police are now carrying out extensive enquiries into the attacks, including door-to-doors and a review of CCTV.
The first incident happened between 10-11 pm in the Dicks Hill area of the town when a 33-year-old man was attacked and knocked to the ground as he making his way home. Whilst on the ground he was further assaulted, suffering a number of injuries to his face.
It is believed that three men were involved in the attack.
A trio of men were also involved in the second attack on a 63-year-old man who was making his way home along Sheuchan Street around 10 pm. As he neared Foreland Place, he was knocked to the ground and at some stage received an injury to his face.
The three assailants are described as wearing hooded tops with the hoods up.
Both victims were later treated at the Galloway Community Hospital.
Detective Sergeant Garry Mitchell said: “At this early stage we have a limited description of the attackers, other than three males, who may have been wearing hooded style tops with the hoods up.
“This is an unusual type of crime for Stranraer and with the two attacks happening in a similar fashion around the same time there is a strong possibility that they are linked. These attacks have taken place on two men making their way home after a day out and at this stage we are unclear as to what the motive may have been. The consequence of these attacks is that we have two victims who have some serious injuries which have required treatment at hospital.
“I would urge anyone who may have been out in Stranraer town on Sunday evening, and, in particular, in either of the two areas where the attacks took place, to get in touch with us if they saw or heard anything suspicious, or if they can help us identify any of these three men.”
Police can be contacted on 101, giving reference number 0889 2/9.