Members of North Sunderland Parish Council were united in their calls after they were told that Seahouses Primary School was also interested in having them.
The whale bones were in the grounds of the former first school until they went missing in December 2018, later found near North Sunderland Football Club.
Parish councillors want the whale bones to be placed on a grassed area overlooking the harbour on Crewe Street, along with a new additional seating area.
Coun David Donaldson said: “I think the whale bones should be in a public place, not in school grounds where access is restricted.
“It’s something we should be proud of having.
“They have been part of the village for many years and I think they should be in a public place where people can see them.
“We had previously decided the harbour was the most sensible place.
“Tourists can see them, we can see them and there is street lighting.
“Because they were in the school that’s how we had them stolen and we don’t want a repeat of that.”
Coun Alan Macfarlane added: “If they’re at the school they will be forgotten about and no-one will pay any attention to them.”
The harbour idea also has the backing of landowner North Sunderland Harbour Commission.
Coun David Shiel, also a harbour commissioner, suggested an interpretation board could also be installed.
The bones are believed to be those of a Greenland right whale, once a common sight in whaling ports, and are likely to have been brought back by one of the Berwick whalers in the 1820 or 1830s.
They lay in a farm barn in North Sunderland for many years before being erected at Seahouses First School in the early 1930s.
Since they were found dumped and partially damaged next to the gates of North Sunderland Football Club in March 2019, Northumberland County Council has been in discussions with North Sunderland Parish Council and landowners to find a new home.