The impending demolition of the five giant towers has fuelled concern among employees over the future of the facility. Govan has headed off a number of threats over the years, but the redundant structures are due to be removed by its owners Clydeport later this month.
This comes at a time when BAE is considering the future of its entire shipbuilding division, with orders from the Royal Navy drying up.
The company is helping to build two aircraft carriers for the Royal Navy and is holding discussions with the Ministry of Defence over the possibility of orders for two Type 26 frigates.
However, even if the contracts are won, work would not begin until some time after the carriers are finished.
Deputy First Minister and Govan SNP MSP Nicola Sturgeon has urged the company to hold off bringing down the cranes until it has 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 out-rageous if Govan doesn't have a future.
"There is a duty on the company to make clear its 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."
While Govan is operated by BAE, it is owned by docks giant Clydeport. BAE denied there was a link between the future of the dockside cranes, which have not been used for five years, and the yard closing.
A BAE spokesman said the move was at its request, adding: "Clydeport is removing dockside portal cranes which are no longer fit for purpose.
"Work is expected to last 18 months."
When asked if this 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 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 the 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."
It is understood the fixed cranes, despite being put up in 1974, remain serviceable and have been approved by external inspectors, despite an incident when a jib fell off one some years ago.
Insiders hope the Type 26 contracts will come to Scotland whatever the outcome of next year's independence referendum. However, the issue remains politically thorny. One option could be that the Type 26s would be built in Scotstoun, after substantial investment, while Govan is closed by 2016. Sources stressed nothing is finalised.