Boris Johnson also appeared to commit to holding a public inquiry into the matter during an exchange in the Commons this week, although further details are yet to emerge.
Tracy Felstead, from Telford, and Rubbina Shaheen, from Shrewsbury, are waiting to hear if they will be granted leave to appeal their convictions after being jailed over financial shortfalls.
In December the Post Office agreed to pay out £57.75 million in compensation to 550 workers, including Miss Felstead, in an out-of-court settlement.
Miss Felstead, 37, was jailed for six months in 2001 after being convicted of stealing £11,500 while a 19-year-old counter clerk.
Mrs Shaheen, 54, was jailed for 12 months in 2010 over a £40,000 shortfall when she was sub-postmistress of Greenfields Post Office.
They say that the discrepancies were caused by a fault in the Post Office's controversial Horizon database which logs all transactions in Post Office branches.
During Prime Minister's Questions this week, Labour MP for Jarrow Kate Osborne raised the case of constituent Chris Head who had suffered as a result of the case.
"These errors have resulted in bankruptcies, imprisonment and even suicide," she told Mr Johnson.
"Will the Prime Minister today assure Chris and others that he will commit to launching an independent inquiry?"
Mr Johnson replied: "I am indeed aware of the scandal to which the honourable lady alludes and the disaster that has befallen many Post Office workers. I have met some of them myself.
"I am happy to commit to getting to the bottom of the matter in the way that she recommends."
Telford MP Lucy Allan, who represents Miss Felstead, welcomed news that there would be an inquiry.
"There is a ground swell of support building in Parliament for the Government to take steps to help clear the names of Post Office workers wrongly convicted of theft based the Post Office's flawed Horizon computer system," she said.
"We also need to find a legal means for all those convicted on unreliable evidence, the faulty IT system, to be cleared."
Last month Ms Allan raised Miss Felstead's case in the House of Commons, asking for a parliamentary debate on the matter.
The Criminal Cases Review Commission is to meet on March 24 to discuss whether Mrs Shaheen, Miss Felstead and others should be granted leave to appeal their convictions.