Space for new graves is rapidly running out at Higher Cemetery in Heavitree and Exeter City Council is considering allowing families to adopt graves which pre-date 1900.
Families would be then be able to take advantage of the unused space beneath and have their own inscription put on the reverse of the original memorial.
Ian Quance, Exeter's bereavement services and cemeteries manager, said: "The challenge is for us to find a sustainable way to maintain these valuable historical landscapes and allow the public to be fully involved in the process.
"This proposal is very much in embryonic form at present and we appreciate everyone's thoughts on the subject. We'd listen to anyone with an opinion — there has to be a consensus that this sort of thing is acceptable given the sensitivity of the issue."
The proposal would involve the council cleaning and repairing old memorials and providing a history of the grave and those buried within it.
It would then be leased to a family, which would assume financial responsibility for any work they required, like inscriptions, and its ongoing upkeep.
Mr Quance added: "We would have to take an overview on any proposed work to ensure that it was consistent with maintaining the fabric and atmosphere of the old cemetery, much as we do with new memorials anyway.
"We would not, for instance, allow the memorial to be removed from the cemetery."
The choice of grave would be limited to the oldest memorials. Those considered to be of historical interest may be excluded.
Den Perrin, secretary of the Heavitree Local History Society, said: "As long as they don't lose the original inscription I think this is a good idea."
Heavitree city councillor David Morrish said the scheme could work if it was introduced sensitively.
He said: "This does seem a sensible way of trying to manage burial grounds that are becoming derelict.
"It may not be to everyone's taste — but some of these memorials are attractive bits of sculpture."
An adopt a grave scheme is already operating in the City of London Cemetery, and director Ian Hussein said it was a success.
He said: "We introduced it in 2005 and to date 30 families have purchased this kind of grave.
"We are quite pleased with that number as it is not something we have pushed but have simply offered it as an option.
"We call them heritage graves. People seem content as it is often a very impressive position and memorial which otherwise would not be available to people.
"It also helps to conserve cemeteries and their landscape, as well as the original memorials."
To give the council your view on adopting a grave contact bereavement services on 01392 265370 or email bereavement.ser vices@exeter.gov.uk.