The 85-year old, who is said by neighbours to suffer from dementia, was stabbed, beaten and sexually assaulted while her husband was at the shops in Morningside, Edinburgh.
Police Scotland said the woman, who was found with life-threatening injuries on Friday, was said to be in a critical but stable condition in hospital yesterday.
Detective Chief Inspector Keith Hardie said the assault, which is being treated as attempted murder, took place in a short space of time after the woman's husband left the house.
He said the woman was critically ill but stable in hospital after being stabbed repeatedly and possibly being the victim of a blunt force trauma.
He added: "She is also displaying injuries which may be consistent with some form of sexual attack."
As a result of the door-to-door inquiries, officers have identified a witness who saw a male standing in the doorway at about 3.30pm on the Friday afternoon.
DCI Hardie said: "That is clearly a crucial time for us because the victim's husband had left the house at that point, leaving our victim on her own."
Police released a description of a man they want to trace in connection with the attack. He is described as aged 18 to 30, of average build with short dark hair.
Officers in the area are continuing door-to-door inquiries and CCTV footage is being examined.
Mr Hardie said the possibility of rape was "absolutely" being looked into as part of the inquiry.
The victim remained unconscious in Edinburgh Royal Infirmary and police said they have not been able to speak to her.
She was on her own in the house for around an hour and a half and was discovered in the living room by her husband on his return from the shops.
Her family were "shocked and traumatised" by the incident, Mr Hardie said.
He said: "In my experience within the service, I've never really heard of anything to this extent, of this nature; an attack on an elderly woman in the safety of her own house, a vicious attack, and plainly an attempt to take her life.
"I would appeal to people if they have been in the area and have seen anything that's been even very, very slightly suspicious to get in touch, because it might be hugely significant to our ongoing inquiry."
He added: "We're dealing with undoubtedly an attempted murder, there's a potential sexual element to it and we can't rule out the possibility of a theft."
According to a neighbour, the woman had been suffering from dementia for several years.
One said: "It's not been easy for her husband as he is quite shaky too, but I always see him going for the bus. You never see his wife.
"She has had dementia for quite a while and I know it's been pretty hard on the family and her husband."
Another neighbour said: "It's shocking. I know they were having a hard time because the woman has dementia and I think her husband has Parkinson's or something similar. Who would do that to a frail lady?"
A couple, who said they were close friends with their neighbours said: "They've lived here for over 30 years, everyone knows them. They're a lovely couple, we can't believe somebody would do this. It makes you scared to be here on your own or just the two of you."
Chief Inspector Richard Horan moved to reassure residents that extra patrols were being carried out in the area. He said: "People are obviously concerned for the victim, for their neighbour.
"I would say that crimes like this are incredibly rare and we will continue to try and reassure the community using every possible avenue.
"We do have extra police officers out on the street, we are able to attend and respond to any concerns people have, we will continue that as long as it takes as we progress the investigation towards detecting and arresting the perpetrator."