The yards will complete the Royal Navy's super-carrrier contract in 2014
Plans are being drawn up for the possible closure of two navy shipyards after aircraft carrier work ends in 2014, BBC Scotland has learned.
It follows a leaked memo from BVT, the owner of yards at Scotstoun and Govan in Glasgow, and in Portsmouth.
The memo says the Ministry of Defence is willing to pay for thousands of redundancies to scale down Britain's capacity for building warships.
The MoD said negotiations were ongoing and no decisions had been made.
The leaked memo, seen by BBC Scotland, shows BVT Surface Fleet's chief executive Alan Johnston forecasting savings of up to half a billion pounds from the closure of two out of three yards after the contract for two super-carriers is completed in 2014.
Douglas Fraser BBC Scotland business and economy editor
While the shipyards have plenty of work to keep them going until 2014, we are now looking at a strong likelihood that half the Clyde's remaining shipbuilding capacity is going to shut within two years of the super-carrier work being complete.
If Portsmouth is the single site chosen to carry on Britain's specialist naval shipbuilding, then both Clyde yards would be forced to close.
Portsmouth, however, is in a weak position if there are to be closures, as I understand it can't build the size of ships necessary for replacement of the Royal Navy's Type 42 destroyers.
This would be part of an agreement for an exclusive BVT deal with Whitehall to build the Royal Navy's future ships.
In the memo, Mr Johnston writes: "BVT has committed to review its industrial footprint in light of the projected reduction in UK shipbuilding requirements post completion of the CVF (aircraft carrier) programme (current projections show that at the time the MoD requirements could be delivered from a single BVT facility) and MoD has committed to underwrite the necessary closure costs.
"These one-off rationalisation/investment costs are estimated to be between £115m to £165m for redundancies, site closure, environmental clean-up, equipment disposal and asset write-downs. Discussions are under way to agree the specific mechanism by which they will be recovered (e.g. via overheads over an agreed time frame)."
BVT said that the memo was about "worse case scenario planning". A spokesman said: "BVT continues to invest in designs, facilities and skills to secure the long-term future of both its Clyde and Portsmouth facilities.
"BVT continues to win orders both in the UK and overseas and is progressing well with a unique 15 years partnering agreement with the MoD that will further secure that future."
A spokesman for the Ministry of Defence said it had to look at the consequences of reduced demand for navy shipbuilding.
"There will be a need for rationalisation and efficiency measures going forward," he added.
He said nothing had been decided.
Jim Moohan, the chairman of the shipbuilding and engineering unions in Scotland, said the news was disappointing, but the workforce on the Clyde would look at all opportunities.
"We understand the problems in shipbuilding, we understand reduced capacity. I just think the timing could have been better planned," he said.
"There could have been better communication. It is something which could happen, but we will rise to the challenge as we have done in the past.
"I believe we have time on our side to look at the opportunities, to make discussion, create it and be positive going forward."
I will be contacting BVT ... to demand assurances over their commitment to the Clyde yards
Nicola Sturgeon Scottish Deputy First Minister
Scottish Deputy First Minister Nicola Sturgeon said: "These reports will cause deep concern among the workforce and communities on the Clyde.
"I have recently spoken with Alan Johnston, chief executive of BVT, who had assured me the company was working to secure the long-term future of the yards and I will be contacting BVT... to demand assurances over their commitment to the Clyde yards and the long-term future of Scotland's shipbuilding expertise."
But defence analyst Dr Alex Ashbourne said the Clyde yards were "in a better state" than the Portsmouth yard.
She added: "I actually think that the Scottish yards will be better placed at the moment because even though the facility at Portsmouth is considerably more modern, they don't have the capacity and the size of the yards, I think, to build and to look after, for example, the aircraft carrier once it's being refurbished, once it's being fitted out."
Scottish Liberal Democrat leader Tavish Scott MSP said: "Labour must immediately end the uncertainty surrounding the future of naval shipyards on the Clyde.
"They cannot leave hard working men and woman high and dry without knowing their future.
"The government must make an urgent statement on this potentially devastating news for Scotland."
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