A 30-year-old man was attacked at a property in Cunnery Road in the town just before midnight.
Andrew Leslie, aged 50, is now wanted in connection with the incident. He is believed to be in Shropshire and could be sleeping rough.
The victim was seriously injured but has been treated for his injuries and they are not considered life-threatening.
Andrew Leslie is white, 5ft 8ins tall, of a slim build with fair and unkempt hair who speaks with a local accent, police said.
Detective Sergeant Mal Normandin said: “Leslie left the address following the incident and has not been seen since and we are making concerted efforts to trace him at a number of associated addresses and locations.
“We believe that Leslie is still in the Shropshire area and possibly sleeping rough. While we would like the public to report any sightings of him we would ask that he isn’t approached but that you call Shrewsbury CID via 101 immediately please and quote 22/49599/21.
“Alternatively, information can be shared anonymously via Crimestoppers by calling 0800 555 111.”
Police tape remained in place on Monday and police said they believe it to have been an isolated incident.
Detective Inspector Dafydd Jones, of West Mercia Police, said: "Officers were called to a property in Cunnery Road, Church Stretton, just before midnight on Sunday, June 13, following a report of assault.
“A 30-year-old man had received a stab wound and was taken to hospital with serious injuries.
"Investigations into the incident are ongoing and I would like to reassure the public that this was an isolated incident.”
A spokeswoman from West Midlands Ambulance Service said: "We were called to Cunnery Road, Church Stretton at 11.38pm on Sunday June 15 to reports of a stabbing.
"An ambulance, paramedic officer and a MERIT trauma doctor and critical care paramedic (CCP) attended the scene.
"The ambulance crew and paramedic officer arrived to find a man in a serious condition and administered emergency care on scene before leaving for hospital on blue lights.
"The MERIT doctor and CCP intercepted the ambulance and the team jumped on board to administer advanced trauma care to the man for the remainder of the journey to hospital."
The incident came just days after a 26-year-old man was stabbed and killed in Telford.
Peter Cairns was found with serious injuries on a footpath near Stonebridge Close, in Aqueduct, and died in hospital later that evening.
Four youths have appeared in court charged with his murder.
In a tribute on behalf of the family, Mr Cairns' mother said her son had been a gentleman who would always do his best to help other people.