Amid widespread cuts in defence spending, Govan and Scotstoun are locked in a battle with their rival base at Portsmouth following a review ordered by the defence giant, which holds significant UK Government contracts.
It is widely expected only two of the three facilities will survive the run-down.
In the strongest signal yet that BAE Systems will end shipbuilding at a yard, chief executive Nigel Whitehead said plans for a "reduction in footprint" could see "the cessation of manufacturing at one of the sites".
He said in an interview: "We will be making decisions this year, so we have a number of weeks in which to do that."
Any decision has implications for the independence referendum. The cross-party, pro-Union Better Together campaign made extensive use of shipyard workers in a video at their launch.
Shadow Defence Secretary Jim Murphy claimed yesterday that one guaranteed way to lose the yards was for Scotland to go it alone, but the closure of Govan or Scotstoun would blunt his argument.
Equally, Yes Scotland could portray the saving of both Clyde yards as a sign of confidence that naval work would continue after independence.
Portsmouth, which is still seen as most at risk, has 5000 workers, with fewer than half directly involved in shipbuilding.
John Dolan of the GMB union told The Herald: "Although the company has never come out and 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 after we have read the bones of Nigel's statement. It is obviously a worrying time with the last of the Type 45 [frigates] leaving in March, and by this time next year we could be in serious bother when the steelworks come to an end on the carriers."
Written agreements are in place over retaining vital skills, but the Ministry of Defence (MoD) insists it is for BAE to decide on how best to deliver its agreed naval vessels.
Last month also saw the collapse of the proposed merger between BAE Systems and the French-German conglomerate EADS that would have created the world's biggest defence and aerospace group.
BAE Systems said it was in contact with the MoD as it reviews its shipbuilding future, adding: "We continue to work closely with the Ministry of Defence to explore all possible options to determine how best to sustain the capability to deliver complex warships in the UK in the future.
"This work is ongoing and we are committed to keeping our employees and trade unions informed as it progresses."
It is feared there will not be enough work to keep all three sites profitable, with a gap between the completion of work on the Queen Elizabeth-class aircraft carriers and the start of the Royal Navy's Type 26 Global Combat Ship programme.
Mr Murphy said: "This is worrying for everyone involved. 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. The Tory Government plans are adding to the worries, but there is one certainty, which is that the SNP proposals would sink Scottish shipbuilding."
Westminster SNP leader and defence spokesman Angus Robertson MP said: "While it is disappointing that BAE Systems are reportedly considering closing one of their 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."