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MPs on the Treasury Committee mentioned that they had "lost confidence" in TSB boss Paul Pester after the bank's IT meltdown.
Nicky Morgan, who chairs the committee, has referred to as on the TSB board to offer "serious consideration" as to whether his place as chief govt is "sustainable".
Thousands of consumers have been unable to entry their bank accounts following a botched pc switchover final month.
TSB is but to subject a response to the MPs' name.
A letter from Ms Morgan to Richard Meddings, TSB chairman, on behalf of the committee, says the MPs had "lost confidence in [Mr Pester's] ability to provide a full and frank assessment of the problems at TSB, and to deal with them in the best interests of its customers".
"If he continues in his position, this could damage trust not only in TSB, but in the retail banking sector as a whole," the letter mentioned.
Commenting on the correspondence, Ms Morgan mentioned TSB's public communications had usually been complacent and deceptive since its IT issues started.
"This tone has been set from the topby Paul Pesterand whether intentionally or not, he has not been straight with the committee and TSB customers," she mentioned.
"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 name comes after the Financial Conduct Authority mentioned on Wednesday that it was investigating TSB and criticised Mr Pester for giving an "optimistic view" of companies after the meltdown.
On Wednesday Andrew Bailey, FCA chief govt, informed the MPs that TSB bosses have been "in a hole and they have got to get themselves out of that hole".