The Treasury select committee has said it has lost confidence in the chief executive of TSB after the fiasco over its IT system.
Nicky Morgan, the committee's chairman, has written to the chairman of TSB, Richard Meddings, questioning Paul Pester's position at the top of the bank.
It comes after the Financial Conduct Authority revealed it was investigating TSB over its IT failings, which led to nearly two million people being locked out of online banking services.
Andrew Bailey, the regulator's chief executive, said that during his previous appearance before MPs in May Mr Pester had given an "optimistic view" of the state of TSB's services at the time.
Mrs Morgan said: "Since the IT problems at TSB began, its public communications have often been complacent and misleading. This tone has been set from the top – by Paul Pester – and whether intentionally or not, he has not been straight with the committee and TSB customers.
"Dr Pester’s statements that 'everything is running smoothly for the vast majority of our […] customers' and that there will be no barriers’ to customers switching accounts, and his denial that there were problems on TSB’s fraud reporting line, are all examples of this.
"The Treasury Committee, therefore, has lost confidence in Dr Pester’s position as chief executive of TSB, and considers that the TSB board should give serious consideration as to whether his position is sustainable."
Problems with TSB's IT system began in April, when the bank began moving its customer data from a system controlled by its former owner, Lloyds Banking Group, to a new system built by its new owner, the Spanish banking group, Banco Sabadell.
The new system proved unable to cope with the number of people attempting to use it and, weeks later, some TSB customers are still facing problems.
Since then Action Fraud has warned of a sharp increase in fraudsters, spotting an opportunity to take advantage of customers, sending out fake text messages and phishing emails claiming to be from TSB.
In her letter to Mr Meddings, Mrs Morgan said the committee was concerned that if Mr Pester continued in his position it would damage trust not only in TSB, but in the retail banking sector as a whole.
Mr Meddings replied to Mrs Morgan's letter saying he would respond fully "in due course" but he added: "TSB has made good progress in resolving the recent IT issues and TSB is now functioning at, or close to, normal for the majority of TSB customers.
"We recognise that we still have areas where we need to improve performance for our customers and we do not underestimate these remaining issues. The important point is that this progress has been achieved under the leadership of Paul Pester, who continues to have the full support of the TSB board."