Labour MP Ian Davidson, whose Glasgow South West constituency includes BAE's Govan yard, suggested a "break clause" in the event of a Yes vote in next year's referendum, meaning the contract for Type 26 frigates would revert back to the UK Government to be reconsidered.
An announcement is expected soon on the contract for 13 frigates, with each vessel estimated to cost £250m-£350million.
Mr Davidson says the UK Government does not build warships in other countries and he believes the decision cannot wait until after next September's vote.
However, the "break clause" suggestion was described by Deputy First Minister Nicola Sturgeon as "outrageous".
The future of Govan and Scotstoun is dependent on the awarding of the Type 26 frigates, which will be built between 2015 and 2021 and provide work for 1500 people.
The ships will replace the Royal Navy's existing Type 23 frigates.
Mr Davidson said the referendum was a factor in the decision on where to place the contract and pointed out the UK Government had stated it does not build warships in other territories.
He said: "There should be no unnecessary delays in awarding the contract. The MoD does not need to wait for the referendum to issue contract.
"It could place the order with the provision that if Scotland separates it would revert back to the MoD where to place it. There would be a break clause in it.
"Then the SNP needs to be clear about what it would do if the order is taken back."
Shipbuilding has been a political football in the independence campaign with opponents arguing a Yes vote will lead to the closure of the Govan and Scotstoun yards.
However, supporters of independence reject the claim and state the referendum has no bearing on the current contract.
Ms Sturgeon said: "This is an outrageous suggestion and it beggars belief that the local MP would argue such a position.
"The contract should come to the Clyde on merit .
"The Clyde is the best place to build these ships for a variety of reasons, not least the skills of the workforce.
"If Scotland becomes independent it will be for Scottish governments to support our shipbuilding industry, including through our own naval procurement and support for exports.
"But responsibility right now lies with the UK Government and it should provide clarity and security for the Clyde yards sooner rather than later."
Mr Davidson has circulated leaflets in his constituency claiming 'Separation shuts shipyards'.
He is chairman of the Scottish Affairs Committee at Westminster, which earlier this year produced a report with the same title stating: "The Ministry Of Defence will not place an order for the Type 26 new Royal Navy frigates until the constitutional position is clear."
It added: "The Clyde shipyards will not be eligible for UK-restricted orders and will have little prospect of winning export work.
"The needs of any Scottish navy will be insufficient to maintain capacity and any attempts to switch into a short run of submarine building would be prohibitively expensive."
In her Evening Times column today, Ms Sturgeon argues both yards could be sustained in an independent Scotland.
She said: "The bottom line is this: I simply don't accept that, in a maritime nation like Scotland, we shouldn't, or we can't, support both of these shipyards.
"Other countries similar in size to Scotland, like Norway, have shipbuilding industries much bigger than ours."
Politicians and union leaders have called on the UK Government to make the announcement soon to allow BAE and the workers to plan for the future.
It is understood meetings will take place at Westminster this week and an announcement on the contract is expected within the next week.
Speculation is growing the order is likely to come to the Clyde, but the reduction in workload and desire to cuts costs will mean they will be built at Scotstoun, with Govan being closed once current work runs out.
It is understood to be thought more cost-effective to concentrate the workforce in one yard.
While both yards were needed to build the sections of HMS Queen Elizabeth and HMS Prince Of Wales - the 65,000-tonne aircraft carriers that provided work for several years - the scale of the new Type 26 frigates and possible future orders do not require two yards.
BAE is due to announce before the end of the year which one of its three naval shipbuilding yards will close, Govan, Scotstoun or Portsmouth.
Last week the Evening Times reported the start of work to remove five iconic dockside cranes at Govan.
The news led to fears within the workforce that Govan was being prepared for closure. But BAE said the landowner Clydeport was removing the cranes at the request of the shipbuilder.
It denied there was any significance in the timing and that it was unconnected to the impending decision on the future of the three UK yards. A BAE spokesman said the cranes were decommissioned in 2008 and the company used mobile cranes, which were safer and more efficient.