BAE systems is planning to shut Govan, Scotstoun or its yard in Portsmouth due to UK Government budget cuts and falling orders from the Ministry Of Defence.
BAE employs about 2000 workers at Scotstoun, at the former Yarrows yard, and 1500 at Govan.
Work on the two big Government contracts that has kept the Glasgow yards alive ends in the next year. Beyond that, the future is uncertain.
Steel cutting on the aircraft carriers HMS Queen Elizabeth and HMS Prince Of Wales ends late next year and the Type 45 frigate work is complete in March, while the next orders for Type 26 global combat ships has still to be made.
Union leaders said they had been expecting the news of one closure but were hopeful the Clyde yards would survive.
BAE has been engaged in a review of its shipbuilding operation this year and the expectation was one yard would go, which has now been confirmed by the company's boss.
Chief executive Nigel Whitehead said he "anticipated a reduction in footprint and that part of that might actually be the cessation of manufacturing at one of the sites".
Mr Whitehead added: "We continue to work closely with the MoD to explore all possible options to determine how best to sustain the capability to deliver complex warships in the UK in the future.
"We are committed to keeping our employees and trade unions informed as it progresses."
Union leaders and politicians urged the firm to make the announcement as soon as possible.
John Dolan, of the GMB union, stewards' convener at Scotstoun, said: "Although the company has never said this before, with the workload dropping off this is a challenge we knew was coming and one we are prepared to face.
"Our priority is the Clyde and we will be seeking a meeting with the company as soon as possible.
"It is obviously a worrying time with the last of the Type 45s leaving in March. By this time next year we could be in serious bother when the steelworks come to an end on the carriers.
"We will be looking for meetings with the MoD, ministers and local MSPs"
Jamie Webster, trade union convener at Govan, said: "I have emphasised very strongly to the company that the sooner the announcement is made the better. The delay will not help anybody.
"Obviously that would be in the next two to three weeks, before Christmas, so we know exactly where we are in relation to the future – although I am very confident the Clyde will come through this."
The other yard that could close is in Portsmouth, where 1500 are employed.
Labour MPs say the possibility of Scottish independence has put the Clyde yards in danger.
Shadow Defence Secretary Jim Murphy, said: "There must be clarity from the UK Government over the future of these yards and workforces. Scotland has such a proud shipbuilding history and it should be a part of our future as well.
"UK Government plans are adding to the worries, but there is one certainty – the SNP independence proposals would sink Scottish shipbuilding.
"The rest of the UK would become a foreign country to Scotland and the UK Royal Navy has not built a warship in a foreign land in living memory.
"The Royal Navy order book keeps Scottish yards afloat but independence would see orders dry up. Thousands of jobs are at risk by Nationalists' plans."
Ian Davidson, Glasgow South West MP, who constituency includes the Govan yard, said: "If Scotland was to separate from the United Kingdom, then the terms of business would preclude any orders for the Type 26 being placed on the Clyde.
"There is a major difficulty about the timing of any Type 26 orders. These are scheduled to be placed before the date of the referendum and it is likely BAE will play safe and place the order with Portsmouth where there is no danger of separation."
The SNP, however, said the Clyde's past performance stood the yards in good stead to survive any reduction.
SNP defence spokesman Angus Robertson said: "While it is disappointing that BAE Systems is reportedly considering closing one of its UK shipyards, the Scottish yards go into this process in a very strong position.
"Just last week leading defence expert Ian Godden highlighted the strength of Scotland's defence industry because of its industrial and engineering capability. These strengths must be taken into account by BAE and, we hope, it will help it conclude there should be no reduction in manufacturing capacity in Scotland."
A Scottish Government spokesman said: "BAE Systems is an important employer in Scotland.
"We will monitor this situation closely and work with BAE with the aim of ensuring any future plans protect the interests of employees and shipbuilding in Scotland."