The boss of NatWest, which owns Coutts, has admitted a "serious error of judgement" in discussing Nigel Farage's relationship with the private bank.
Dame Alison Rose said she was "wrong" to respond to questions from the BBC about his bank account being closed.
The ex-UKIP leader had demanded NatWest explain how his financial information was made public as the row over his bank account closure escalated.
NatWest said it still had full confidence in Dame Alison at the helm.
Dame Alison's apology comes after the BBC apologised on Monday for its inaccurate report earlier this month which said Mr Farage no longer met the wealth threshold for Coutts, citing a source familiar with the matter.
Mr Farage later secured a Coutts report which indicated his political views were considered.
Dame Alison said in conversations with BBC business editor Simon Jack "believing it was public knowledge" she had confirmed that Mr Farage was a Coutts customer and he been offered a NatWest bank account.
The NatWest boss said she did not reveal any personal financial information about Mr Farage.
"In response to a general question about eligibility criteria required to bank with Coutts and NatWest I said that guidance on both was publicly available on their websites.
"In doing so, I recognise that I left Mr Jack with the impression that the decision to close Mr Farage's accounts was solely a commercial one," she added.
When Coutts decided to close Mr Farage's account, he said it did not give him a reason.
Mr Farage subsequently obtained a document looking at his suitability as a Coutts customer.
The 40-page document flagged concerns that he was "xenophobic and racist", and also questioned the reputational risk of having Mr Farage as a client.
It said that to have Mr Farage as a customer was not consistent with Coutts' "position as an inclusive organisation" given his "publicly stated views".
Dame Alison said she had not been involved in the decision to close Mr Farage's account, but Coutts had told her it was for commercial reasons.
She said when she spoke to Mr Jack, she had not seen the dossier released by Mr Farage.
"I was wrong to respond to any question raised by the BBC about this case. I want to extend my sincere apologies to Mr Farage for the personal hurt this has caused him and I have written to him today," she added.