Rishi Sunak has declined to say whether he has confidence in the BBC chairman, saying he cannot speculate while an inquiry is held into his appointment.
Richard Sharp is under scrutiny after it emerged he had acted as a go-between for a loan guarantee for then-prime minister Boris Johnson
An MPs' committee has said Mr Sharp made "significant errors of judgement" in doing this while applying for the BBC job.
He insists he got the job on merit.
Questioned on Monday, Mr Sunak said he would not "pre-judge" the outcome of an investigation by the government's appointments watchdog.
Asked whether Mr Sharp had undermined the impartiality of the BBC, Mr Sunak said the controversy over his appointment related "to a process that happened before I was prime minister".
Mr Sunak said he couldn't "speculate" or "prejudge 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 spokesperson twice repeated No 10 was "confident" in the process but refused to expand further, citing the ongoing investigation.
At the fourth time of questioning, when asked "does Richard Sharp retain the Prime Minister's support?", the spokesperson replied: "Yes, again I don't have much more beyond what I have already said."
Pressure is growing on the BBC chairman after a critical report by MPs found he showed "significant errors of judgement" in acting as a go-between for Sam Blyth, a Canadian millionaire and distant cousin of Mr Johnson
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.
The Digital, Culture, Media and Sport Committee published a critical report on Sunday, concluding he should not have become involved in the facilitation of a loan while applying for the BBC job.
It 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 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 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.
Veteran BBC broadcaster Jonathan Dimbleby told BBC 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".
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.