Potential suppliers have been asked to get involved in "market engagement" for the contract for three Fleet Solid Support (FSS) ships, which is expected to be worth more than £1 billion.
The offer is open to "UK and international suppliers" – prompting a furious response from former defence minister Mr Spellar, who wants all MoD contracts to go to British firms.
He accused ministers of having "sneaked" the announcement out during parliamentary recess in a bid to avoid scrutiny, and said the move went against the British shipbuilding industry.
Branding the decision "an absolute disgrace", the Labour MP said: "The Defence Select Committee will undoubtedly be taking this up – which is presumably why they have sneaked this out in the recess.
"And spare us any bleating excuses about EU procurement rules; just buy and build British."
The ships are being built to help keep the Navy's new aircraft carriers at sea by providing them with storage for ammunition and food.
The MoD says the market engagement process will help work out the contract before negotiations start regarding the construction of the 40,000-tonne vessels.
General secretary of the Shipbuilding and Engineering Union, Ian Waddell, has said the project could provide a vital boost to Britain's post-Covid economic recovery.
"The FSS programme is a perfect example of what the Prime Minister would call a shovel-ready project," he said.
"It will get the post-coronavirus economy up and running as part of the levelling up agenda by injecting £1.3bn into regional economies and could benefit every single shipyard in the UK."
The building of the supply ships could help fulfill Sir John Parker’s national shipbuilding strategy review, which has called for ships to be built in domestic shipyards on British soil.