The SS Nomadic, the last of the White Star ships and a vital link to the Titanic story, is due to be auctioned.
The Nomadic took passengers out to the Titanic
The luxury ferry, built by Harland & Wolff in Belfast, is to go under the hammer in France on Thursday afternoon.
The ship, built the year before Titanic, took first class passengers to the great liner which sank with the loss of more than 1,500 people in 1912.
Campaigners have been lobbying the government in an attempt to take the ship back to Belfast.
Belfast Sinn Fein councillor Michael Browne said he was concerned at the cost of such an enterprise.
"I don't think it is a council responsibility to act alone here and attempt bring the Nomadic back to Belfast," he said.
"From what I understand the SS Nomadic is worth little more than scrap value."
The Alliance assembly member Sean Neeson, who represents Northern Ireland on the UK Historic Ships Committee, said the ship could help develop the city's links with the Titanic.
"We're part of an island race," he said.
"The very fact that we have had perhaps the most important shipbuilding yard in the world and have depended so much on the sea over the years, I think means there is an urgency.
"I would hope at the very last minute the Northern Ireland Office would step in and put forward a bid," he said.
However, the Department of Social Development has, as yet, refused to be specific on whether it will commit to plans to save the ship.
The reserve price on the Nomadic is £165,000.
Following a public appeal, £40,000 has been raised in private pledges and Belfast City Council has agreed to contribute £100,000.
But even if the ship is secured, much more money is needed to bring her back to Belfast from France intact.
The 221st ship ferried passengers to the White Star liner Olympic, and in April 1912, it did the same job for Titanic.
Nomadic saw service in both world wars and was later used as a restaurant on the Seine in Paris.
More recently, it has been languishing semi-derelict in the port of Le Havre.
A feasibility study by Belfast City Council estimated the cost of buying the ship and bringing it back to the city would be around £750,000.
It would then need about £7m to restore the ship to its former glory.
Campaigners, including Belfast Industrial Heritage, have been behind efforts to bring Nomadic back to the city where it was made.
It is hoped that the ship will become the centrepiece of a new tourist quarter dedicated to the world's most famous ship.
Other attractions include the slipway where Titanic was built, the drawing offices where the blueprints for the ship were drawn and the Thompson Dock and pump house where she was fitted out.
Titanic entered into legend in 1912 when more than 1,500 people died during its maiden voyage from the UK to America.