Postal Services Minister, Kevin Hollinrake, the MP for Thirsk and Malton, told the Commons he expected the Government to unveil a solution shortly, after holding talks with Justice Secretary Alex Chalk on Monday.
The Government was facing fresh calls to act after an ITV drama returned the huge miscarriage of justice relating to the Horizon IT system to the spotlight.
Mr Hollinrake told the Commons that he had a “very positive meeting” with Mr Chalk as they discussed how to help the convicted branch managers clear their names.
With more than 700 subpostmasters having received criminal convictions for allegations such as theft and false accounting, Mr Hollinrake criticised the “slow pace” of them being overturned.
“We have devised some options for resolving the outstanding criminal convictions with much more pace,” the minister said.
“While the scale of the problem is immense, the Government is unwavering in its resolve to tackle it, to compensate those affected and to leave no stone unturned in the pursuit of justice.”
Mr Chalk will first speak to senior figures in the judiciary, Mr Hollinrake said, but he expected ministers to announce the proposals “very shortly”.
Mr Hollinrake said that “proper and thoughtful consideration” will also be given on how private prosecutions are undertaken “to make sure a scandal like this can never happen again”.
But he said that they had discussed “at length” the possibility of bringing a Bill to Parliament to quash the convictions.
Mr Hollinrake said that “full and final” compensation has already been paid to 64% of those affected.
“Getting justice for the victims of this scandal and ensuring that such a tragedy can never happen again is my highest priority as a minister and has been throughout my 15 months in office,” he added.
Former High Court judge Sir Gary Hickinbottom will chair an independent panel assessing the financial losses of postmasters with overturned convictions, it was announced.
A petition to the Forfeiture Committee calling for Ms Vennells, who ran the Post Office while it routinely denied there was a problem with its Horizon IT system, to lose her CBE has attracted more than one million signatures.
Mr Sunak’s official spokesman said: “The Prime Minister shares the public’s feeling of outrage on this issue. He would strongly support the Forfeiture Committee if it chose to review the case.”
Mr Hollinrake has said it is “perfectly reasonable” to ask the former Post Office boss to hand back her CBE.
He also suggested that Fujitsu, the firm behind the faulty accounting software that made it look like money was missing from shops, and anyone else shown to be responsible should be “held accountable including making any payments” into the compensation fund.
Sir Keir Starmer has called for prosecution powers to be stripped from the Post Office and previous convictions looked at again.
The Labour leader said: “I think that the prosecution should be taken out of the hands of the Post Office and given to the Crown Prosecution Service.
“And these convictions, the remaining convictions, need to be looked at en masse.”
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 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.
