Justice Secretary Alex Chalk has met with senior judges to discuss possible solutions to the Post Office IT scandal.
The meeting focused on speeding up the process for sub-postmasters who want their convictions overturned.
More than 700 people received criminal convictions after the Post Office introduced faulty software.
Post Office minister Kevin Hollinrake said the government was investigating ways to overturn convictions.
Mr Hollinrake suggested this could include possible legislation.
He also said that Fujitsu – the technology company behind the faulty software – and anyone else found to be responsible should be “held accountable including making any payments” towards victims’ compensation.
“We have devised some options for resolving the outstanding criminal convictions with much more pace,” the minister said.
The scandal has been described as one of the biggest miscarriages of justice the UK has ever seen. Between 1999 and 2015, the Post Office prosecuted 736 sub-postmasters and sub-postmistresses – an average of one a week – based on information from a computer system called Horizon.
Many maintained their innocence and said they had repeatedly reported problems with the software.
But some went to prison for false accounting and theft. Many were financially ruined.
The scandal was the subject of an ITV drama broadcast last week that has catapulted it back into public consciousness.
Alan Bates, a former sub-postmaster who has led efforts for justice, told the BBC he believed a resolution was now “closer”.
Ms Vennells, who was Post Office chief executive between 2012 and 2019, has said she will hand back her CBE with immediate effect.
It comes after a petition calling for her to do so reached more than a million signatures on Monday.
