The plane will be part of a £16m Bristol Aerospace Centre and museum at the former Filton airfield in neighbouring South Gloucestershire.
Labour and Conservative general election candidates for Filton and Bradley Stoke called on George Ferguson and Bristol City Council to invest in the project.
The BBC has invited Mr Ferguson to comment.
The comments were made during a general election debate on BBC Radio Bristol.
'Very appropriate'
Labour's Ian Boulton said the museum would benefit the whole region.
"There's a real danger that this will become the South Gloucestershire aero collection so we'd like to get Bristol on board with that," he said.
Conservative Jack Lopresti said: "I still haven't given up hope of getting a meeting with George Ferguson to see if Bristol are willing to help with the project."
The Liberal Democrat candidate Pete Bruce, said it would be "very appropriate" for Bristol to contribute towards the cost of the museum.
"It will be good for the area as a whole and good to have Bristol as a partner helping with the exhibits and renovations," said Mr Bruce.
Last week, a £2m Budget windfall towards the cost of the centre was announced.
The airfield site includes two Grade II-listed hangars built by the Royal Flying Corp during World War One.
They were later home to 501 Squadron, which won seven honours in World War Two and was one of the most heavily-engaged units in RAF Fighter Command.
Bristol's mayor has been urged to help pay for a permanent home for Concorde.