Hilda Marcel died in 1987 from lung failure and her husband Eugenios died from prostate cancer seven years later.
Their mummified bodies were kept in a basement flat in the city's Gilmore Place, which was regularly visited by a son of the couple, until 2002.
Since their discovery, the bodies have been stored in the city's mortuary.
Edinburgh city council has been trying to get the family's consent to bury Mr and Mrs Marcel, but this has been repeatedly refused.
Now, almost 30 years after Mrs Marcel died, the council has lodged a petition at the Court of Session to seek permission to dispose of their bodies.
Prof Roderick Paisley from Aberdeen University's Law School said: "The wheels of justice do grind slowly, and they appear to be grinding incredibly slowly on this particular hearse.
"I think this should have been dealt with three or four years ago.
"I do appreciate there are sensitivities here, but the law is reasonably clear."
A police investigation was carried out when the bodies were found in 2002, but no charges were brought.
Death certificates had been correctly issued and the pair had died of natural causes.
Cameron Rose, leader of the Conservative group on Edinburgh City Council, said it was an "extraordinary story".
He added that there needed to be "sensitivity for the families".
The Court of Session has now agreed to hear the case, although so far no date has been set.
A council has launched a legal action in an attempt to bury the mummified bodies of a husband and wife who died decades ago in Edinburgh.