Chancellor Jeremy Hunt has “significant concerns” about the conduct of NatWest boss Dame Alison Rose, BBC News has been told.
Dame Alison admitted a “serious error” in talking about Nigel Farage’s relationship with its private banking arm Coutts.
She said she had been wrong to respond to questions from the BBC about Mr Farage’s account being closed.
Dame Alison has faced calls to resign from Mr Farage and several Tory MPs.
Those calling for her resignation include former cabinet minister David Davis.
Nigel Farage, the former leader of UKIP, 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 on Tuesday afternoon comes after the BBC apologised for its inaccurate report earlier this month which said Mr Farage’s account was being closed because he no longer met the wealth threshold for Coutts, citing a source familiar with the matter.
In her first admission that she had been involved, Dame Alison said in conversations with BBC business editor Simon Jack “she had confirmed that Mr Farage was a Coutts customer and he had been offered a NatWest bank account”.
She said she had believed this was public knowledge.
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 said.
She added: “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.”
Simon Clarke, MP for Middlesbrough South and East Cleveland, tweeted that the chancellor would be right to have concerns.
“The whole issue has been a disgrace from start to finish,” he wrote.
MP Saqib Bhatti, the Conservative Party’s vice-chairman for business, earlier said: “While it’s not for politicians to determine what the company should do, her position would appear to now be untenable.”
