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Richard Sharp is facing renewed pressure to stand down as BBC chairman.
Jonathan Dimbleby and Baroness Wheatcroft have joined a growing chorus of voices questioning his position.
Mr Sharp is facing criticism for his role in facilitating a £800,000 loan for then-prime minister Boris Johnson.
An MPs' committee said Mr Sharp made "significant errors of judgement" in doing so while applying for the BBC job. He insists he got the job on merit.
Veteran BBC broadcaster Mr Dimbleby told the BBC's Newsnight: "What [Mr Sharp] should do honourably is fall on his sword."
He warned the credibility of the corporation in the public's view was at stake, adding "the BBC needs this like it needs a hole in the head".
Crossbench peer Baroness Wheatcroft, who sits on the Lords Communications and Digital Committee, said it was "impossible" not to agree with Mr Dimbleby.
"Even if Mr Sharp behaved absolutely correctly, it doesn't look right, it doesn't smell right, and it doesn't feel right for the BBC to have a chairman who is now being questioned about his judgment," she told BBC Radio 4's Today programme.
"It may be the sort of thing that happens all the time in the circles that Mr Sharp moves in, and it may be that £800,000 is just chicken feed as far as he is concerned, but to most people who love the BBC £800,000 is a massive sum," she added.
"He did a favour for a prime minister who was in need at a time when the prime minister was being asked to do Mr Sharp a massive favour and grant him one of the plum jobs in British broadcasting."
BBC News has contacted Mr Sharp and all other BBC board members for comment.
When questioned on Monday, Rishi Sunak declined to say whether he had confidence in the BBC chairman, saying he would not "pre-judge" the outcome of an inquiry by the independent office for public appointments, which he said would determine whether "rules and procedures were adhered to".
Later, asked directly if Mr Sunak had confidence in Mr Sharp, the prime minister's official spokesperson said: "Yes, we are confident the process was followed.
"This was a two-stage process, including assessment by an advisory assessment panel, constituted according to the public appointments code.
"But there is a review into this process and we will look at that carefully."
The Digital, Culture, Media and Sport Committee published a report on Sunday, concluding Mr Sharp should not have become involved in the facilitation of a loan while applying for the BBC job.
Mr Sharp acted as a go-between for Sam Blyth, a Canadian millionaire and distant cousin of Mr Johnson.
The report found that Mr Sharp should have disclosed his knowledge of the talks when asked to provide a written account of his potential conflicts of interest during his BBC application.
The report's authors urged him to "consider the impact his omissions will have on trust in him, the BBC and the public appointments process".
Mr Blyth had said he was willing to act as guarantor on a loan reportedly worth up to £800,000 for the then-PM after reading media reports he was in financial difficulty.
Mr Sharp, who was working as a Treasury adviser at the time, approached Simon Case, the country's most senior civil servant, to arrange a meeting between the pair.
At the time he had already applied for the BBC job and was advised to have no further involvement in the talks.
Mr Sharp insists his involvement in the matter ended with that single meeting, despite admitting he met socially with Mr Johnson and Mr Blyth at Chequers months later.
Last week he told MPs he "didn't arrange the loan" but did not refute acting as a "sort of introduction agency". He also described himself as a "go-between" for Mr Blyth and the Cabinet Office.
He admitted the affair had embarrassed the BBC but insisted he had "acted in good faith to ensure that the rules were followed".
"As a go-between I was not between Mr Blyth and Mr Johnson, but I was actually seeking to ensure that due process was followed by ensuring that Mr Blyth had contact with the Cabinet Office before he would do anything to help his cousin," he added.
The chairman is in charge of upholding and protecting the BBC's independence and ensuring the BBC fulfils its mission to inform, educate and entertain, among other things.
Under the terms of the BBC Charter, the government must hold a "fair and open competition" to find a suitable candidate.
Once ministers have chosen a preferred candidate from the applicants, the prospective chairman has to submit themselves for questioning by a parliamentary select committee.
The culture secretary can formally dismiss them from the post following consultation with the rest of the BBC's governing board if they decide they are "unable, unfit or unwilling" to perform their duty.