The bodies of George Garcia, 68, and wife Betsy, 65, were discovered in a top floor flat within Crathes Court, Muirend.
It is believed Mrs Garcia took her own life after an incident that saw her husband killed.
Police are investigating the tragedy, but say they are not looking for anyone else in connection with the deaths.
A post mortem will be carried out to determine precisely how they died.
Last night a uniformed police officer stood guard outside the housing complex, which has 52 one and two bedroom flats.
Plain clothed officers were seen walking in and out of the building, while forensic teams were investigating inside.
It was believed the couple may have been trying to reconcile after a temporary split.
Harold Hamilton, 77, has lived in the flat next door to the Garcias since December. He said a care worker had called at the couple's home on Friday but there was no answer.
He said: "The care worker was trying to get buzzed into their flat. He told me there was no answer, but that he had seen them in the morning.
"I haven't seen either of them for about two weeks.
"The last time I saw them was when I was doing my laundry in the communal room.
"I thought it was a bit odd that all the lights were off and their bedroom blind was closed on Saturday night, usually there are a few lights on."
The couple were found about 9.30am yesterday.
Mr Hamilton added: "Mrs Garcia was quite talkative and always chatted away to me, but he never said a word to me."
Another neighbour, who did not want to be named, added: "I have never seen any family visit them.
"I think they had been married, separated and then got back together recently."
Other neighbours spoke of their shock at the deaths.
Richard Cornes, who lives in a ground floor apartment, said: "I saw about four police vans, paramedics and CID officers arrive.
"I knew something awful had happened. This is such a quiet neighbourhood, you would never expect anything like this to happen.
Margaret McTaggart, 78, has lived in the complex for 21 years. She said: "I am shocked to find something so horrible might have happened here. It's the first thing to have happened here the whole time I have been here."
Jenny Daalman, 31, who lives across the road from the complex, said: "I saw all these police cars and ambulances and couldn't believe it. This neighbourhood is so quiet, so everyone will be in shock."
Crathes Court, which is managed by property management company Grant & Wilson, has staff on site and a community alarm service. It is for people over the age of 55.
A police spokeswoman said: "Inquiries are continuing and a post mortem will be carried out to establish the cause of their deaths."