Officers said they are treating the attack as attempted murder and the possibility of rape on the Edinburgh woman was "absolutely" being looked into.
The woman, who was found with life-threatening injuries in her house in the capital's upmarket Morningside area, is described as being in a critical but stable condition at Edinburgh Royal Infirmary.
Police said they had advised residents not to open their doors to strangers.
They are searching for a man described as between 5ft 10in and 6ft and of slim to medium build seen by residents in the area at the time of the attack.
The man, aged between 18 and 30, has dark hair and was wearing a light coloured top and dark trousers, and may have been wearing a baseball cap.
He is also described as having "bushy eyebrows".
The woman was attacked last Friday between 3pm and 5pm at her house on Morningside Grove while her husband had left the house to visit the shops.
Police said she was stabbed "numerous" times and displayed "injuries which may be consistent with some form of sexual attack." Officers revisited the scene of the crime yesterday.
Police have been carrying out high-visibility patrols in the area to offer reassurance to locals.
Local residents say they have been told by police not to answer the door to any strangers. Police confirmed this but Detective Chief Inspector Keith Hardie, who is in charge of the investigation, said: "That's not particular to this area - it's routine crime prevention advice."
He added: "This was a brutal assault on an elderly lady and we are treating it as attempted murder.
"We have been examining CCTV in the area and from the buses that pass through, and by revisiting the area we hope we can jog people's memory who may have noticed something suspicious around Morningside Grove last Friday.
"We are keen to trace a man who was seen standing near the locus around 3.30pm on Friday afternoon [July 11].
"He is described as 18 to 30 years old, average height and build, and with short dark hair.
"This is a crucial time as this is when the victim was alone in the house."