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.

The RMT revealed it had received a legal challenge just hours before the start of a three-day walkout.
Southern said in a statement it was "considering all possible options to stop the strike and that includes a legal challenge".
Talks aimed at averting the strikes over changes to the role of conductors ended without agreement last week.
The RMT said it intended to continue with the planned action but was examining the paperwork.
In a message to union members, RMT general secretary Mick Cash said: "The dispute is not over and the strike action... is definitely on.
"Organise the picket lines and protests and continue to show your determination and solidarity in your fight for justice."
The strike is due to begin from one minute past midnight on Tuesday, causing fresh travel misery for passengers.
In talks last week, the RMT said the train operator had rejected a fresh set of proposals that could have resolved the dispute.
But Govia Thameslink CEO Charles Horton said the RMT counter-proposal "didn't come anywhere near our goals of modernising our train service for the benefit of passengers".
RMT general secretary Mick Cash said: "RMT can confirm the receipt of a formal legal challenge from Southern Rail in the guards safety dispute.
"It is appalling that rather than sitting down with us at Acas today to seek a resolution the company has chosen to run to the courts under the cloak of the anti-union laws."
On Sunday, it emerged rail staff had been advised by union bosses to accept the new contracts.
The RMT said there was "no climb down" but it was their duty to protect staff against mass sackings.
A Southern spokesman said: "On Friday they told their conductor members to accept a deal and then tomorrow they plan to strike against it.
"It is a situation which will leave our passengers baffled and in that context we are presently considering all possible options to stop the strike.
"We have written to the RMT about the validity of the dispute in light of recent developments.
"We await their response and will consider all possible options to stop the strike and that includes a legal challenge."
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.
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Southern is taking legal action to try to stop a series of strikes from Tuesday, the RMT transport union said.