He made "significant errors of judgement" acting as a go-between on a loan for Boris Johnson, while applying for the post, the committee said.
The SNP's John Nicolson, a committee member, told the BBC Mr Sharp's position was "extremely difficult".
Mr Sharp said he did not help arrange a guarantee on the loan or give Mr Johnson financial advice.
Speaking on BBC One's Sunday with Laura Kuenssberg, Labour's Lisa Nandy also said Mr Sharp's position was becoming "increasingly untenable".
Mr Sharp's involvement in the then-Prime Minister Mr Johnson obtaining an £800,000 loan guarantee has come under scrutiny since the Sunday Times first reported the claims last month.
Businessman Sam Blyth, a distant cousin of Mr Johnson and Mr Sharp's friend, had reportedly raised the idea of acting as a loan guarantor for Mr Johnson in 2020.
Mr Sharp was named as the government's preferred candidate for the BBC chairmanship in January 2021 and at the time the Commons Digital, Culture, Media and Sport (DCMS) Committee backed his appointment.
The government's choice is ultimately decided by the prime minister, on the advice of the culture secretary, who is in turn advised by a panel.
This week the BBC chairman was recalled to appear before the committee and its report was published on Sunday.
Speaking to MPs he said he had introduced his friend Mr Blyth to the Cabinet Office.
In its report, the cross-party committee criticised Mr Sharp's failure to mention any involvement he had in events surrounding the loan when they were considering his suitability for the job two years ago.
The report said his decisions to "become involved in the facilitation of a loan to the then-prime minister while at the same time applying for a job that was in that same person's gift" and failure to disclose this to the committee undermined confidence in the public appointments process.
The MPs concluded: "Mr Sharp should consider the impact his omissions will have on trust in him, the BBC and the public appointments process."
The report, carefully worded, does not say in black and white that he should quit. But it says he should "consider the impact" of what has happened, which is a diplomatic way of raising that point.
On Sunday With Laura Kuenssberg, the SNP's Mr Nicolson went further than the report had, saying Mr Sharp's position was "extremely difficult" after he "broke the rules".
He said: "He has lost the trust of the BBC staff, that's very clear.
"When you sign up for that job application you are asked if there's anything about your relationships with anybody that could cause embarrassment.
"This has clearly caused embarrassment."
Mr Nicolson said Mr Sharp had not told MPs at the time of his appointment that he had facilitated an £800,000 loan for Mr Johnson "who then gave him the job".
"It's all a bit banana republic," he added.
Cabinet minister Andrew Mitchell told the same programme it was up to the BBC to decide what to do over Mr Sharp's future. He also said it was important to wait for a review into his hiring by the watchdog that oversees public appointments.
But he was challenged on this during the programme as according to the BBC charter the chairman can only be removed from post by the government, not the BBC.
Shadow levelling up secretary Lisa Nandy said it was "right that the BBC is able to publish" its own review into Mr Sharp's appointment, and that "due process is followed".
But she added that when the BBC's report was published, if questions still could not be answered, "the integrity of the BBC is more important than one individual".
This week Mr Sharp told the committee he had met Cabinet Secretary Simon Case in December 2020 to get permission to pass on businessman Mr Blyth's details to him.
However, at the same meeting he had told Mr Case that he had applied for the BBC job, and therefore agreed he would have "no further participation" in order to avoid any conflict of interest or perception of conflict given his application to the BBC.
In its report, the DCMS Committee said Mr Sharp had recognised the need to be "open and transparent" by bringing it to the attention of the cabinet secretary, but "failed to apply the same standards of openness and candour in his decision not to divulge this information during the interview process or to this committee during the pre-appointment hearing [for the BBC job]".
"Mr Sharp's failure to disclose his actions to the panel and the committee, although he believed this to be completely proper, constitute a breach of the standards expected of individuals applying for such public appointments," the report added.
A spokesperson for Mr Sharp said he did not facilitate an introduction between Mr Johnson and Mr Blyth and he was not involved in the arrangement of a loan between them.
"Mr Sharp appreciates that there was information that the committee felt that it should have been made aware of in his pre-appointment hearing. He regrets this and apologises.
"It was in seeking at the time to ensure that the rules were followed, and in the belief that this had been achieved, that Mr Sharp acted in good faith in the way he did."
The spokesperson added: "Mr Sharp believed he had dealt with the issue by proactively briefing the cabinet secretary that he was applying for the role of BBC chair, and therefore beyond connecting Mr Blyth with Mr Case, he recused himself from the matter."
The DCMS Committee report was also critical of ministers who had defended the decision to endorse Mr Sharp in 2021 after the row over the loan broke, despite the fact they had not been told about the situation.
"The fact that ministers have cited this committee's original report on Mr Sharp's appointment as a defence of the process was followed, when we were not in full possession of all the facts that we should have had before us in order to come to our judgement, is highly unsatisfactory," the report said.
Prime Minister Rishi Sunak has said Mr Sharp was made BBC chairman following two "transparent and rigorous" appointment processes, adding the appointment was made by his predecessor.
According to the Sunday Times, a leaked memo from Mr Case allegedly warned Mr Johnson to "no longer" ask for financial advice from Mr Sharp.
But the MPs in this new report said there was an "unresolved issue" as to why the cabinet secretary had believed Mr Sharp had been giving financial advice to Mr Johnson and called on the Cabinet Office to "clear up the confusion relating to the advice given to the prime minister immediately", given that Mr Sharp had said this was not the case.
A Cabinet Office spokesperson said: "We do not comment on leaks."
The Office of the Commissioner for Public Appointments is still looking into the appointment process of Richard Sharp.
The BBC is also conducting its own internal review over any potential conflicts of interest Mr Sharp may have in his role as BBC chairman.
It is not known when ether of these reviews will be concluded.
John Nicolson, SNP MP: BBC chairman's position "is extremely difficult"
Labour's Lisa Nandy: BBC chairman's position "increasingly untenable"