TSB chief executive Paul Pester told MPs: "I am deeply sorry to say that the issues we created as a outcome of migration were an opportunity for criminals to target TSB customers".
Nicky Morgan, chair of the Committee, said: "I am deeply concerned by TSB's poor communications about the scale and nature of the problems it has faced; by its response to customer fraud; and by the quality and accuracy of the oral and written evidence provided by Dr Pester".
"T$3 he current communications were perceived as poor, and could reduce trust in TSB and in the banking sector as a whole". "I ask that the board consider the committee's view as a matter of urgency".
During a planned upgrade to TSB's systems in April, there were widespread reports of internet and mobile banking being down after the scheduled downtime came to an end.
It went on to say that parliament was "troubled by the complaints and compensation process, which is not meeting the high standards that you have personally promised the committee".
While around 12,500 customers have requested to leave TSB since the outage, 370 customers who wanted to switch banks faced a software glitch, which led to their regular payments being cancelled because the bank had told companies they had died.
"In this regard, the tone has been set from the top, by your chief executive".
Mr Pester apologised repeatedly as he appeared before the committee on Wednesday after a hard first hearing on May 2 during which chair Nicky Morgan described his evidence as "the most staggering example of a chief executive who seems unwilling to realise the scale of the problem that is being faced".
In his response, Mr Meddings said Mr Pester "continues to have the full support of the TSB board".
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Mr Pester has "not been straight" with customers at the bank since its online services were plunged into chaos, it said.
He said: "We recognise that we still have areas where we need to improve performance for our customers, and we do not underestimate these remaining issues".
Mrs Morgan said: "Since the IT problems at TSB began, its public communications have often been complacent and misleading".
"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".
In her letter to Meddings, Morgan listed instances where the TSB boss had either omitted information or made statements the committee deemed inaccurate, and did not take opportunities to correct his statements.
"This is absolutely incendiary stuff", he said.
The head of the Financial Conduct Authority (FCA) said in a letter to Treasury Committee chairwoman Nicky Morgan that the regulator would carry out a joint investigation with the Prudential Regulation Authority.
Mr Bailey also appeared before MPs ahead of TSB to give evidence into the TSB debacle.