The Justice Secretary was holding crunch talks with judges on Tuesday on expediting clearing the names of hundreds of subpostmasters who were wrongfully convicted in the Horizon scandal.

Work and Pensions Secretary Mel Stride, touring broadcast studios on Tuesday morning, said the discussions are taking place as the Government considers ways to overturn the convictions, including possible legislation.

Ministers “intend to move very quickly” to resolve the issue, he said, after the miscarriage of justice was brought into the spotlight by an ITV drama.

Mr Stride told Times Radio: “Alex Chalk, the Justice Secretary, is in discussions with senior people within the justice system, judges and so on, to look at what the options are to make sure that we push through that backlog of appeals as quickly as possible.”

On Sky News, he said: “My understanding is that they are happening now.

“So this is something that is happening hour by hour. It’s not something that’s going to happen next week. It is happening right now and we intend to move very quickly.”

A solution could be announced by the end of the week, the Cabinet minister said.

With more than 700 subpostmasters having received criminal convictions for allegations such as theft and false accounting after the Post Office introduced faulty software, Mr Stride was challenged over the slow pace of them being cleared and whether the Government was asleep at the wheel.

“I wouldn’t accept that,” he said, pointing to the public inquiry examining what went wrong being set up some years ago and the payout of £138 million of compensation.

Asked whether Rishi Sunak as chancellor was slow to pay out compensation to subpostmasters, Mr Stride said: “No, I don’t think that is a fair charge at all.”

MPs have called for Fujitsu, the firm behind the faulty Horizon accounting software that made it look as if money was missing from shops, to pay for compensating wronged Post Office staff.

Mr Stride suggested the company might have to stump up if the inquiry concludes it blundered, saying it “won’t necessarily just be the taxpayer” who is “on the hook for this money”.

He told LBC Radio: “To the extent that that culpability rests upon the shoulders of others than Government, then I think you can expect ministers to come to the appropriate conclusions. And perhaps it won’t be just the taxpayer that is on the hook for those costs.”

But he stopped short of saying the Japanese tech firm should be barred from being awarded millions of pounds of Government contracts while the Post Office inquiry is ongoing.

“My view is that we need to wait to see what the inquiry decides in terms of culpability.

“Now, in the event that it determines that Fujitsu made a number of knowing mistakes and caused all sorts of problems that wouldn’t have otherwise have occurred, then that would strike me as being quite a serious situation and I would expect some very serious consequences.

“But we don’t know that at this stage until the inquiry goes through all its due processes.”

Ministers are looking at changing the rules around private prosecutions by companies, Mr Stride said, amid calls including from Labour leader Sir Keir Starmer to strip prosecution powers from the Post Office.

The senior Tory also said Liberal Democrat leader Sir Ed Davey has “big questions” to answer on his role in the Horizon scandal as postal affairs minister between 2010 and 2012.

Reports suggest that 50 new potential victims have approached lawyers since ITV’s Mr Bates vs The Post Office was broadcast.

The Post Office is wholly owned by the Government.