The bodies of George Garcia, 68, and his wife Betsy, 65, were discovered in a top-floor flat within Crathes Court in Muirend, Glasgow, at around 9.30am yesterday.
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 examination will be carried out to discover how the couple died.
Last night a uniformed police officer stood guard outside the quiet housing complex, which is a modern development of about 50 one and two-bedroom flats near Glasgow Hutchesons' Aloysians rugby club.
Plain-clothed officers were seen walking in and out of the building as forensic teams began their investigations.
It is understood the couple may have been trying a reconciliation after a temporary split.
Harold Hamilton, 77, has lived in the flat next door to where the Garcias lived since December.
He said a care worker had called at the couple's home on Friday but there was no answer.
He said: "On Friday there was a care worker trying to get buzzed in to 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."
Mr Hamilton said he was shocked by their deaths.
He said: "They were both quiet, he was more quiet than her, he never said a word to me."
Another neighbour, who did not want to be named, added: "I've never seen any family visit them. They are so quiet.
"I think they had been married, separated and then got back together recently."
Crathes Court, which is managed by Grant Wilson, provides wardens on site and is for people over the age of 55.
Neighbours spoke of their shock at the double deaths.
Richard Cornes, who lives in a ground-floor apartment in the block, said: "I saw about four police vans, paramedics and CID officers arrive in the morning.
"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: "This is a very quiet place to live and I'm very happy living here.
"I'm quite shocked to find that something so horrible might have happened here."
Jenny Daalman, 31, who lives across the road from the complex, said: "This morning I saw all these police cars and ambulance and I couldn't believe it.
"This neighbourhood is so quiet.
"It is absolutely mad to know something like this has happened.
"The whole neighbourhood will be in complete shock."
A Strathclyde Police spokeswoman said: "Inquiries are continuing and a post-mortem will be carried out to establish the cause of their deaths."