The man and woman found dead in a house in Heiton on Boxing Day have been named locally as Kevin Armstrong, 53, and Pauline Cockburn, 48.
However, police have not yet confirmed their identities as formal identification of the bodies has yet to take place.
It has been reported that Mr Armstrong killed his partner before shooting himself, but that has not been confirmed by police.
The police did confirm this morning that no one else is being sought in connection with the two deaths, though.
Mr Armstrong’s brother Tommy, of Kelso, took to social media to express his sadness over the tragedy.
He wrote: “Not the Christmas I expected, in shock still. After a horrible incident, two people have lost their lives.
“I don’t know what happened, but I do know there are kids out there that have lost their mummy and dad, and we should be thinking of them and their families at this sad time.
“Rest in peace to you both.”
Pauline, who on her Facebook page describes herself as a carer at SB Cares, is thought to have had three children.
One of them, her son Paul Gresham, was reported in daily newspapers as posting an angry message on Mr Armstrong’s page after the deaths, saying ‘Kevin Armstrong, rot in hell.’
He has since deleted his post.
The couple’s bodies were discovered in the house in Ladyrig View, Heiton, at around 3pm on Tuesday, December 26.
Detective Chief Inspector Keith Hardie is leading the investigation.
He said: “The deaths remain unexplained pending post-mortem examinations being carried out.
“Whilst work to establish the exact circumstances is ongoing, I can confirm that the incident is contained and I am not looking for anyone else in connection with the deaths.
“We continue to offer our thoughts and support to the families of those who have died and to the local community, who I’d like to thank for their assistance with the investigation and for the support they in turn are providing to the families and each other.”
Superintendent Jim Royan added: “This is a small, close-knit community on the outskirts of Kelso, and we recognise the impact that the deaths and our ongoing investigation is having.
“Whilst work is ongoing at the house in Ladyrig View, local officers will remain in the area.”
Anyone with information that could assist the inquiry should contact the major investigation team on 101, quoting incident 2012 of December 26, or Crimestoppers anonymously on 0800 555111.