John Spellar accused the Ministry of Defence of using European Union regulations as an "excuse" to not commit to the call.
Foreign companies have been invited to compete for the contract for the three Fleet Solid Support (FSS) ships – worth more than £1 billion.
It prompted a backlash from the former defence minister and Warley MP who wants all MoD contracts to go to British firms.
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.
Mr Spellar, in a question to Defence Secretary Ben Wallace, said: "Back in July, he said 'such ships are not highly complex, so once the competition happens I do not think it would take too long to build them'.
"He continued: 'British shipbuilding and British yards produce some of the best ships in the world, we should support them as best as we can and ensure our navy gets some British-made kit.''
"As EU regulations are no longer the excuse - if they ever really were - why won't the Secretary of State commit today not only to build these ships in British yards, but also to get a move on?"
Ben Wallace said he was "keen" to ensure the Government made the right decision – and the right decision for the Royal Navy.
He said: "[John Spellar MP] will know that first of all, one of the challenges in our yards is not that they can't make ships.
"It is simply that there is a feast and famine. Sometimes we go from a pipeline that is full, to a pipeline that is empty and it is incredibly important that we make sure we schedule our ship building to make sure we keep as much productivity going and throughput into our yards as possible.
"On the point of the Fleet Solid Support ships, we have started discussions and a competition will be issued. He will know that previous competition was stopped.
"I'm keen to get it right, I'm keen to make sure we get it right for our Royal Navy and I would say to him he should wait for the competition to be issued."
The MoD said the market engagement process will help work out the contract before negotiations start regarding the construction of the 40,000-tonne vessels.