The GMB has written to the First Minister inviting her to debate the issue in front of staff from Faslane, Coulport, Rosyth and the upper Clyde shipyards.
Gary Smith, Scottish secretary of the union, said the scrapping of Trident would put jobs and communities at risk.
In a letter to the First Minister, he said: “GMB represents many workers employed in the defence industry, including those who work on the Clyde and at Rosyth.
“Those working women and men and their families are counting on a future of full employment, they know that the only guarantee for that is for Trident to be replaced by a new nuclear defence deterrent stationed in Scotland.
“Scotland has deep-seated inequality, high levels of unemployment, underfunded public services, an underdeveloped industrial base, a housing crisis; these should be key election issues rather than Trident replacement.
“However, given the decision you have taken to raise Trident’s successor as a Scottish Parliament election issue, I would like to challenge you to debate this question in front of an audience of working women and men from the upper Clyde and Rosyth at a venue, time and place of your choosing.”
Ms Sturgeon described Trident as “immoral” and “impractical” during an anti-nuclear rally in London’s Trafalgar Square at the weekend.