Sacking Gavin Williamson as Defence Secretary will save two years and hundreds of millions of pounds during the restoration of Parliament, insiders have claimed.
Plans to move MPs out of Parliament to nearby Richmond House got a shot in the arm when Mr Williamson was removed from office last week.
The former Defence Secretary was locked in a bitter dispute with Commons' authorities about handing over the Ministry of Defence car park to help those managing the project.
The Mirror understands that negotiations with Mr Williamson's successor Penny Mordaunt were going much better and it is estimated that could save the restoration project hundreds of millions of pounds.
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It came as the government introduced a bill in the wake of the fire in Notre Dame to set out the multi-billion pound bid to restore the Palace of Westminster.
The bill would see a delivery body set up to manage the project, modelled on the 2012 London Olympic authority.
A Sponsor Body, made up of MPs, Peers and external members would oversee the project - being accountable to Parliament for spending and delivering the project.
An Estimates Commission made up of two MPs and two Lords will scrutinise spending plans on the project, which would see MPs decant to nearby Richmond House in the mid-2020s to let the project work begin.
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Leader of the House of Commons Andrea 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 restored for future generations.
"The bill ensures the vital work needed to protect its future will happen in the most efficient way."
The Bill requires the sponsor body to secure parliamentary support for the design, cost and timing of works as well as to "consult widely" on the plans.
But Labour's Lord David Blunkett said the government is showing a "lacklustre" approach to the restorations of Parliament.
The Former Home Secretary said that the plan would mean that they could progress without winning public support - damaging the proposals.
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He said: "The contrast between the urgency and decisiveness displayed in France with the lacklustre approach of the Government here is palpable."
Since 2008 there have been 66 dangerous fires in Parliament, despite 24/7 fire patrols inside the building.
The bill is expected to be approved by MPs in the Autumn, with MPs moving to a replacement chamber and office complex in the old office of Department of Health, which will be completely renovated for the project.
Early estimates suggested the bill for the project would be £3.5billion - but insiders have suggested the total cost is expected to rise.
MPs backed a motion to support the restoration by all moving out during a heated debate in January last year.
