Post Office minister Kevin Hollinrake is to meet Justice Secretary Alex Chalk this afternoon (Monday).
Mr Hollinrake, MP for Thirsk and Malton, is expected to discuss how to help the convicted branch managers clear their names.
The Horizon scandal is expected to be raised in the Commons as Parliament returns this week amid growing anger over the wrongful prosecution of subpostmasters and postmistresses for alleged fraud.
Conservative backbencher David Davis and Labour MP Kevan Jones are pushing for an emergency debate after an ITV drama returned the widespread miscarriage of justice to the spotlight.
It comes as calls grow for former Post Office chief executive Paula Vennells to hand back her CBE and measures to be introduced to secure redress for the victims.
Former Tory minister Mr Davis stressed it was down to Commons Speaker Sir Lindsay Hoyle to grant urgent questions from MPs but he believed there was "no doubt" the issue would come up in the chamber in the coming days.
"This is such a big issue. There are now tens of millions of people who care about this, and care about it a lot," he told BBC Radio 4's Today programme.
"It did take a docudrama in this case I'm afraid, and many of us have been struggling for a long while to try and elevate it."
More than 700 Post Office branch managers were given criminal convictions after faulty Fujitsu accounting software called Horizon made it appear as though money was missing from their shops.
Treasury minister Bim Afolami said on Tuesday Justice Secretary Alex Chalk was "urgently" looking at how to accelerate the "slow wheels of justice" and warned that senior Post Office executives from the period will need to account for their actions.
"(Senior Post Office executives) absolutely need to account for their actions because it will not just be Paula Vennells," he told ITV's Good Morning Britain.
"The head of the justice system ... is urgently looking right now to see what we can do to accelerate these slow wheels of justice so that justice is served for (the victims), and know that this Government is on their side and wants to sort it out."
Prime Minister Rishi Sunak and Labour leader Sir Keir Starmer last week resisted backing calls for the former Post Office chief to lose her CBE.
But shadow environment secretary Steve Reed said on Monday the honour is an "insult" to Horizon scandal victims and should be handed back as a petition calling for Ms Vennells to be stripped of the CBE attracted more than one million signatures.
"In terms of the honour that the former head of the Post Office received ... well, speaking for myself personally here, I do think she should give that back," he told TalkTV.
"It's an insult to the subpostmasters who suffered from what her leadership delivered that she should accept that honour."
Ms Vennells, who ran the Post Office while it routinely denied there was a problem with its Horizon IT system, has said she was "truly sorry" for the "suffering" caused to subpostmasters who were wrongly convicted of offences.
Lord Arbuthnot, who was an MP at the time of the scandal, said Parliament should work to see that the miscarriage of justice is dealt with in a "mass way".
"I think that this is a mass problem which has got to be dealt with in a mass way, and I think that Parliament has got to step in and say that Post Office convictions are not safe," he told Times Radio.
"We've seen the way that investigators behaved and they behaved in a way which was contrary to the rule of law. These convictions have got to be set aside."
Meanwhile, Labour MP Mr Jones, who is a member of the Horizon compensation advisory board, said a "number of individuals" should be prosecuted over the scandal.
"(There are) umpteen charges that could be laid against a number of individuals. That has got to happen," he said.
Another member of the board predicted people would be charged over the scandal as Scotland Yard probe potential fraud offences arising out of the initial prosecutions.
"I think that is what we would normally see in a case like this. I think public pressure and the level of investigation, the inquiry, the police and the professional regulators, I think, are all on the case," Professor Richard Moorhead told Times Radio.
Mr Chalk is expected to meet Post Office minister Kevin Hollinrake on Monday to discuss how to help the convicted branch managers clear their names.
There has been fresh public backlash to the scandal after ITV aired a drama about the scandal last week starring actor Toby Jones.
Reports suggest since Mr Bates Vs The Post Office was broadcast, 50 new potential victims have approached lawyers.
The Post Office is wholly owned by the Government and a public inquiry into Horizon is ongoing.
Scotland Yard said on Friday that officers are "investigating potential fraud offences arising out of these prosecutions", for example "monies recovered from subpostmasters as a result of prosecutions or civil actions".
The Metropolitan Police had already been looking into potential offences of perjury and perverting the course of justice in relation to investigations and prosecutions carried out by the Post Office.
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