The bodies of a man in his 50s and a woman in her 40s were discovered at a property in Ladyrig View in the village of Heiton, near Kelso, at around 3pm on Boxing Day.
Investigations are continuing, with forensic officers examining the scene on Wednesday.
Police said the deaths are currently being treated as unexplained.
Detective Chief Inspector Keith Hardie of Police Scotland's major investigation team said: "I would like to offer my condolences to the families of the man and woman who have sadly died.
"Whilst our inquiries are at an early stage and formal identification is yet to take place, we are satisfied we know who they are and we are providing support to their loved ones.
"Officers and forensic experts will be on site today carrying out a detailed investigation. The deaths are being treated as unexplained at this stage."
Superintendent Jim Royan appealed for anyone with information to contact officers.
He said: "We understand that this is a shocking incident for a small, quiet area of the Borders and we will have a number of our local officers on patrol to provide support to the community as well as the investigation.
"I would like to thank the people of Heiton in advance for their patience and cooperation, and please come and speak to one of the team if you have any information that could assist."
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