A £4bn restoration of the crumbling Palace of Westminster could be kicked into the long grass after the government said MPs would get a vote this month on whether the cost of the repairs could be justified.

Andrea Leadsom, the leader of the House of Commons, said MPs would be asked to vote on a motion authorising the repairs to go ahead but with a review before the end of 2022 on whether there was a need for comprehensive works.

If MPs fail to agree that motion, they will be given a chance to decide on whether to vote for the issue to be examined by an independent body over the next two years.

“The house needs to decide whether we can afford to justify the work that undoubtedly needs to take place to restore this palace – a Unesco world heritage site with over 1 million visitors a year – at a time when there are great fiscal constraints,” she said.

“It is a genuinely open decision that the house needs to make, and what the government has sought to do, taking into account the broad range of views across the house on what should happen, is to put forward, first, an open discussion about whether the house is willing to bear the cost from the taxpayer’s purse.

“Secondly, if the house does believe that now is the time, we need to think about how can we go about doing these things to ensure the very best value for taxpayers’ money. That is incredibly important.”

It is likely to be a free vote, making the result unpredictable, but a number of MPs are unhappy that the government is refusing to allow amendments to the motions, seemingly in an attempt to minimise changes to the proposals that could involve speeding them up.

A joint committee on the Palace of Westminster has already looked into the issue in depth and advised that comprehensive works are necessary, potentially involving MPs and peers moving out of the building for about six years. It said work on removing asbestos, rewiring, replacing plumbing and many other repairs were essential, and backed a 2014 study by Deloitte that estimated the “full decant” option to cost between £3.5bn and £3.9bn.

The Labour MP Chris Bryant, who sat on the committee, said the vast majority of opinion polls showed parliament was “one of the most loved buildings in Britain, it’s a honeytrap for tourists and it preaches our values of democracy to the world”.

“They are fighting shadows and putting it all in peril,” he said. “We are going to vote on one motion and if that does not succeed we will vote on the other. The idea that it is unamendable will not last. They are trying to be too clever by half and normally that fails. Decent Tory MPs will join opposition parties in saying: ‘Up with this we will not put.’”