It comes after the National Audit Office said the scope of the £533 million Future Fit scheme risks having to be changed as funding has only been secured for £312m of that amount.
Shrewsbury and Atcham MP Daniel Kawczynski, who had called for the department to investigate the cost to the taxpayer of delays to the project, met Mr Sunak on Tuesday to discuss the issues.
In a letter, which he presented to the Chancellor, Mr Kawczynski called on him to release the necessary funding for the scheme as part of the Government's autumn comprehensive spending review.
He told him the "people of Shropshire urgently need the project to begin to see an improvement in their hospital services".
The Future Fit project, which has since been renamed the "hospitals transformation programme", will see Royal Shrewsbury Hospital house the county's main A&E.
The 24-hour emergency unit at Telford's Princess Royal Hospital will be downgraded to a part-time 'A & E local' centre and consultant-led women-and-children services will move to Shrewsbury, leaving PRH as the county's site for planned care.
In his letter to the Chancellor, Mr Kawczynski writes: "A budget of £312m has been approved and allocated to date.
"However, more concerningly, there appears to be a shortfall of £221m related to an anticipated £134 million in increased costs for the core elements of the programme, caused by inflation; £24 million to ensure the development meets all current requirements for space and design; and other expenses amounting to a further £63m.
"If additional funding is not secured, the scope of the proposal may have to be changed which could require further consultation and could affect the timetable for works completing.
"We appear potentially to be locked in a cycle of delay and increasing costs.
"Of note that support for the programme from clinicians has not wavered.
"I am writing to ask you to confirm that you will release the necessary funding as part of the autumn comprehensive spending review.
"It is vital that the public of Shropshire begin to see some progress being made in this programme.
"A start needs to be made to demonstrate that large infrastructure projects can be commenced within the timeframe of the next comprehensive spending review."
Mr Kawczynski added that the meeting was "positive' and Mr Sunak said he would evaluate his concerns and would write back to him.
He added: "What we can't afford is for this project not to be implemented because it's now having an impact on waiting times and standards of care."
It comes as the Shropshire, Telford & Wrekin Defend Our NHS group has made fresh calls for the scheme to be scrapped, calling it 'ludicrous'.
In a statement, the group says: "The local NHS is skint and making deep cuts over the next five years.
"And no, they don’t have £221 million stuffed down the back of the sofa.
"That extra money isn’t there – and even if it was, it would be much better spent in developing community services that provided care closer to home, kept people as well as possible and stopped people going to hospital in the first place.
"This disastrous project has been lurching from crisis to crisis since November 2013.
"It was pretty daft then. It’s just plain ludicrous now. Bankrupting the local NHS in order to smash up existing hospital care is as stupid as it gets.
"It’s time to pull the plug on Future Fit. It’s beyond a joke now."
Health bosses say the Future Fit scheme will improve patient care and help reduce cancellations of planned procedures.
But it has been met with widespread opposition.
The plans were reviewed under the instruction of Health Secretary Matt Hancock in 2019, following a request from Telford & Wrekin Council, after being confirmed by the local clinical commissioning groups in January of that year.
Scores of protestors turned up to the Future Fit decision-making meeting at Harper Adams University, near Newport, with some holding up banners which said the county should keep both its A&Es.
In a recent letter to Mr Kawczynski, comptroller and auditor general Gareth Davies, from the NAO, said it is anticipated that construction work will begin in 2023 and finish in 2028, although he warns "any change in scope could require further consultation" which would affect the timetable.
He said if additional funding is not secured, the proposal may have to be "descoped" to make it affordable within the £312 million funding approved to date.
The Shrewsbury and Telford Hospital NHS Trust (SaTH), which runs Princess Royal Hospital in Telford and Royal Shrewsbury Hospital, said the initial business case for the project is expected to be complete by the end of July.
The project will separate planned and emergency care in the county, although PRH will get an 'A&E local'.
Chris Preston, interim director of strategy and planning at SaTH, said: “The changes to local health and care services envisaged as part of the Future Fit consultation are an essential part of improving the health and wellbeing, and meeting the future needs, of the communities across Shropshire, Telford and Wrekin.
“A programme has been established to deliver these crucial changes, with key objectives focusing on safer care and better outcomes for patients, bringing our services and care closer to people’s homes, and increasing preventative activities to keep people well.
"Other benefits include streamlining the care and services people receive, reducing cancellations of planned care, and reducing the amount of time people stay in hospital, which should all contribute to an improvement in patient experience.”