BAE Systems, which owns Govan and Scotstoun shipyards on the upper Clyde, is reported to be considering closing its Portsmouth dockyard.
The defence giant said a review was under way, but did not comment on the report and whether the Clyde yards would benefit by having work transferred from the south.
Work on the hull sections of the Royal Navy's new aircraft carriers HMS Queen Elizabeth and HMS Prince of Wales is currently taking place at Portsmouth and on the Clyde.
Around 1500 jobs would be threatened at Portsmouth and Ian Waddell, national officer of Unite, said the news was "deeply unsettling" for those who worked in the yard and for the thousands of others who depended on it.
He said: "BAE must meet the unions as a matter of urgency to discuss all solutions for saving the workforce. Equally, the Government has a role to play.
"Its cuts are ultimately the reason Portsmouth's docks are now having to fight for a future."
The BAE spokeswoman said: "As part of our business planning activity, we are reviewing how best to retain the capability to deliver and support complex warships in the UK in the future.
"We will keep our employees and trade union representatives fully informed as it progresses."
SNP defence spokesman Angus Robertson said there had been "gross mis-management" of the UK defence budget by Labour during its time in office and it was a "great shame" that shipbuilding workers in Portsmouth and Scotland were concerned about their jobs as a result of that and subsequent cuts by the Tory Government.
Shadow Defence spokesman Jim Murphy said: "It's the size of the Navy order book that sustains the yards in Scotland.
"But as the unions have warned, it's crystal clear that if we leave Britain then we leave the Royal Navy."
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