Isabelle Sanders, 51, was murdered and Norman Busby, 85, left badly wounded after the man confronted them in their home in Raeswood Drive in the Crookston area of Glasgow.
Police said the man had tried to steal a BMW car at shops close to the scene of the killing more than two hours earlier.
They said he had then wielded a “sharp object” as he confronted the couple at about 2am on Wednesday.
Mr Busby, a retired policeman, remains in a high dependency unit at the city’s Southern General Hospital, officers said last night.
Detective Superintendent Jim Kerr, from the Major Investigation Team, said police were following a positive line of inquiry.
He said: “At the Spar branch 700 yards away, a 33-year-old man was assaulted by a man matching the same description and an attempt was made to steal the man’s BMW.
“The man failed to steal the car and was last seen running off in the direction of the park. We do not know what happened between the incident at the Spar and the incident here.
“Norman was in his house watching television with lights on when the incident happened and the suspect gained entry through the front door. This has been a sustained attack on Norman and Isabelle and we would expect there to be some form of blood on clothing.”
DS Kerr appealed for witnesses to come forward.
He added: “We would appeal for a taxi driver who was seen in the area to come forward and for a driver to come forward who had to take evasive action to avoid hitting him.
“The driver of the BMW was taken to hospital and treated for superficial injuries but was released that night.
“There were two cars removed from the property and we are interested in the Vauxhall Astra that was removed for forensic investigation.”
Yesterday a neighbour who rushed to the couple’s aid said he had found Mr Busby badly injured.
The man said: “I think they’ve come to the door and forced their way in and done it. I heard the alarm go off and was alerted.
“He was in a bad way when I saw him, he had been seriously assaulted, but he was talking away at the time.”
Police arrived at the pair’s semi-detached home on Wednesday morning following reports of an incident.
Ms Sanders was discovered inside and pronounced dead, while Mr Busby was treated for injuries and taken to hospital.
The father-of-three is helping police with their inquiries and a post-mortem examination is being conducted to establish how his partner died. Officers were still searching their property and the surrounding area yesterday.
Neighbour Annette Haggerty, 33, laid flowers at the scene in memory of Ms Sanders.
She said: “I used to see her on the bus at night on the way home from her work.”
Further tributes to Ms Sanders were made by work colleagues from Lloyds Banking Group. A spokeswoman said: “Isabelle will be sorely missed by many colleagues who worked with her during her time with the group. We offer our deepest condolences to family and friends.”
Police described the suspect as white, Scottish, 20-30 years old, and 5ft 10-11in tall with a slim build.