Gavin Williamson’s sacking as defence secretary could save the taxpayer hundreds of millions of pounds by speeding up the restoration of parliament, senior sources have suggested.

Williamson had been resisting plans to decant MPs to a replica Commons chamber in Whitehall to allow crucial renovations at the Palace of Westminster because the work would involve the potential use of a Ministry of Defence car park next to its headquarters.

A report by a joint parliamentary committee on the restoration accused the MoD of refusing to engage with the plans and warned that working around the car park could add £350m to the cost and delay the work for around three years to 2028.

Sources close to Williamson, who was sacked last week over a leak from the National Security Council which he denies, earlier this year hit back at the suggestion that he was blocking the moves to protect his VIP car parking spot, telling the Mail on Sunday there were “extremely compelling security reasons”.

But a senior parliamentary source said that “constructive discussions” have now been held on the car park issue, which was now more around bicycle storage than “nuclear bunkers”, and that the delay be shortened to less than a year.

The source said: “We were extremely concerned that the MoD car park appeared to be incredibly precious but it’s becoming less precious as the days pass.

“One of the key questions now is ‘what do we do with the bike racks’, which I think takes us from ‘what shall we do from the nuclear bunkers’ – we are improving.

“We are hopeful that we will find a way there.”

It came as Commons Leader Andrea Leadsom prepared on Wednesday to introduce legislation designed to speed up efforts to protect the Palace of Westminster from a catastrophic event like the recent Notre Dame fire.

The parliamentary buildings (restoration and renewal) bill will be tabled before MPs today, as sources warned that the Palace was deteriorating faster than it can be fixed and that there had been 66 small fire incidents in the building since 2008.

Interim measures including rolling 24/7 fire patrols are already in place to protect the building and ensure it complies with the safety laws, and officials warned the risks of a catastrophic failure within the building are “very great” and that urgent mechanical and engineering work is “vital”.

The bill establishes a 2012 Olympics-style sponsor body of MPs, peers and members of the public to oversee the delivery of the works and a delivery authority to keep costs of the renewal – last estimated at around £4bn – within budget.

Under current plans, the historic Palace will be emptied in the mid-2020s with MPs moving to a new replica Commons in Richmond House on Whitehall, and peers to a replica Lords in the nearby Queen Elizabeth II centre, potentially for more than five years.

Leadsom said: “Events like the terrible fire at Notre Dame bring home to us sharply the importance of preserving our historic buildings.

“The Palace of Westminster, recognised the world over as a symbol of democracy, must be protected for future generations.”

The legislation, which will also include financial safeguards, will ensure the right expertise and “proper structures” are in place to “make sure we deliver the best possible value for taxpayers’ money”, Leadsom added.