Police Scotland said the bodies were found at a house in Troon Avenue.
The remains have not been formally identified but Ms Burke’s relatives have been informed.
Detective Superintendent Graeme Mackie said: “This is a significant development in what has been a particularly challenging inquiry for everyone involved.
“Bennylyn’s family have been informed of this development and we will continue to support them at this difficult time as our investigation continues.”
The 25-year-old mother and two children were reported missing from their south Gloucestershire home on March 1, having been last seen on February 17.
A second child has been found and is being supported, according to police.
Andrew Innes, 50, from Dundee, appeared at the city’s sheriff court on March 8 accused of murder in connection with the two deaths.
He made no plea and was remanded in custody.