More than 95% of Unite members at the Faslane nuclear submarine base and Coulport armaments depot voted to reject a 1% pay deal offered by engineering firm Babcock.
They say it would be "unequal treatment" with colleagues at Devonport, Plymouth and Rosyth in Fife who have received pay awards of between 4% and 5%.
Unite is planning mass meetings with its 800 members early next week before confirming a precise schedule for strike action.
It is expected any action will be in mid-February, as the union has to provide seven days' notice.
Workers who have backed the strike range from electricians, engineers and staff who monitor radiation to cleaning staff. The union says it is the first industrial-side strike for 42 years.
Britain's Vanguard-class submarines is based on the Clyde. The subs are based at Faslane while their Trident missiles and warheads are stored and loaded at the nearby Coulport base.
Unite says Babcock management has angered the workforce by claiming the imposition of a 1% pay rise and £500 lump sum is a result of its contract with the Ministry of Defence, and an expectation that Clyde must mirror the UK government's 1% public-sector pay cap.
But the union says colleagues in Rosyth, where hundreds of workers are currently assembling HMS Queen Elizabeth, the biggest warship ever ordered by the Royal Navy, have been offered a 5% pay rise and a £1200 lump sum.
The union says workers in Devonport have been offered 4.5% and a £1150 lump sum.
Unite national officer Ian Waddell said: "Workers on the Clyde deserve better than the below-inflation pay cut Babcock wants to impose on them and this overwhelming mandate for strike action is a reflection of their anger.
"This is a conscientious, highly skilled collective that understands the unique challenges of their working environment and have consistently worked with the employer down the years to help deliver sustainability for both Babcock and the MoD on the Clyde.
"It is shameful that Babcock is trying to hide behind the MoD and say its hands are tied by the contractual arrangements."
In November, Babcock said work on the new 65,000-tonne Queen Elizabeth Class aircraft carrier helped lift its revenues by 9% to £1.7 billion in the six months to the end of September.
The engineering firm's pre-tax profits for the period grew 17% £141.7 million and Babcock said its £12bn order book gave it confidence.
Mr Waddell added: "Put simply, we cannot accept regressive public-sector pay policy being imposed on private-sector workers and Babcock Marine should stand up for the loyal workforce.
"It's not too late to reach a negotiated settlement - and we are ready to do this with the company directly or at Acas."
Babcock would not comment on the union warning or its claim of unfair treatment. A spokeswoman said: "Discussions on pay negotiations are a matter for Babcock and our employees and we continue to try to find a way forward."
Arthur West, chairman of the Scottish Campaign for Nuclear Disarmament, said he was concerned the dispute may the be result of a "creeping privatisation" of the British nuclear programme.
"The role which Babcock Marine play at Faslane has grown and the number of civilians directly employed by the Ministry of Defence has declined," he said.