The GMB has said future work building new frigates and support ships for the Royal Navy could be awarded to Appledore, with a ferry contract keeping it open until then.
GMB official Matt Roberts said: "GMB feels the Scilly Isles plan could be a realistic way to save the shipyard and prevent the devastation its closure will cause to our community.
"If the Government can't give Babcock the assurances the company needs to take this project on, it must take ownership of this historic yard to protect our sovereign shipbuilding capability post-Brexit."
On Tuesday, former First Sea Lord urged people to "keep fighting" to save the shipyard from closure.
Speaking at Westminster, Lord West warned the loss of Appledore in Devon would be "highly destructive" for the community and also said that other yards were under threat unless there was a "core" amount of shipbuilding work.
Raising his concerns with the Government, he said: "Appledore shipyard has been amazingly competitive in producing survey ships for the Navy, parts of Type 45 and parts of the carrier, and it won the latest competition for four Irish OPVs (offshore patrol vessels).
"It is a very efficient and very capable yard. I find it extraordinary that a firm has to close it because it has problems on its balance sheet elsewhere in its structure."
"We are paying for fleet solid support ships to go elsewhere out of the UK, but that is an opportunity to add to the loading, enabling us to keep key shipyard workers and these key industries going.
"We must keep fighting to keep it open."
Responding, Defence Minister Earl Howe said: "I do not think it is a fair charge if his implication was that the Government have not been supporting our yards at home."
He pointed out Babcock alone has had £1.7 billion of business in the last year.
He added: "As regards the fleet solid support ships, there is no bias on the part of the Ministry of Defence. This is an open competition and we encourage British shipyards to bid."