The five structures -the last of their kind on the Clyde - are expected to come down this month.
Their removal has sent a tremor of concern through the business, despite the fact they haven't been used for five years.
BAE Systems is currently reconsidering the future of shipbuilding division as orders from the Royal Navy dry up.
Today Deputy First Minister Nicola Sturgeon urged the company to hold off on bringing down the cranes until it had finalised its plans.
She said: "I am hugely concerned about the future of Govan Shipyard.
"There's mounting speculation in the yard that it doesn't have a future.
"BAE has to come clean. The workforce has been through the mill too many times before.
"It would be outrageous if Govan doesn't have a future.
"There is a duty on the company to make clear their intentions.
"I would hope the decision to take down the cranes would be put on hold until a decision on the future of the yards is made. I don't want to see Govan shipyard close."
The company denied any link between the cranes coming down and the future of the yard, which employs 1500 people.
A spokesman for BAE said: "Clydeport is removing dockside portal cranes which are no longer fit for purpose."
He admitted it was at the request of BAE.
The spokesman added: "There are five dockside cranes which haven't been used since 2008, which are being removed as they are no longer fit for purpose.
"Work is expected to last 18 months."
When asked if it was connected to the impending decision on closure of one of the three BAE yards, he said: "These are separate issues. This is the removal of decommissioned cranes.
"There is no reason for the timing. There is no significance to the timing."
BAE said it had replaced the old fixed cranes with five mobile ones shared between Govan and its other Clyde yard, Scotstoun.
The spokesman said: "Mobile cranes are safer and more efficient for the construction of our complex warships."
A source familiar with the yard acknowledged mobile cranes could do the work of fixed ones.
But he added: "Their removal is hugely symbolic."
"Why remove them now when there is no need to do so?
"Why emasculate the shipyard? These iconic structures are part of the Glasgow skyline.
"These are berth cranes - the last on the Clyde actually used to build ships."
The Evening Times understands the fixed cranes, despite being put up in 1974, remain serviceable and have been approved by external inspectors.
BAE Systems, currently helping to build two new aircraft carriers, is in talks on orders for Type 26 frigates for the Royal Navy. Work on these wouldn't begin until some time after the carriers are finished.
Sources are hopeful this work will come to Scotland, whatever happens in next year's referendum, although they remain a major political football.
One option, insiders say, is that the Type 25's would be built in Scotstoun, after substantial investment, while Govan would be closed by 2016. But nothing is finalised, sources stressed.