Mr Sharp's appointment is under review following successive reports in the Sunday Times that he helped Mr Johnson secure a loan before getting his job.
The paper's latest story cites a leaked Cabinet Office memo from December 2020. Officials have declined to comment.
Both men deny any wrongdoing.
They have separately said that Mr Sharp was not involved in arranging a loan for the then-prime minister, and say that neither man acted with any conflict of interest.
In the wake of the Sunday Times's fresh report, a spokesman for Mr Johnson insisted to the BBC that he had never received or sought financial advice from Mr Sharp.
Mr Sharp has resisted calls to stand down from his job - saying he expects to be exonerated, as he was appointed "on merit".
BBC News has been told Mr Sharp's position remained unchanged following the latest Sunday Times reporting.
However, his appointment is to be investigated by the public appointments commissioner as well as an internal panel. He also faces a grilling from MPs.
The Conservative government has also faced attacks from opposition parties over the appointment of Mr Sharp, whose role as BBC chairman is to uphold the broadcaster's independence.
Mr Sharp has previously acknowledged that he had contacted Cabinet Secretary Simon Case in late 2020 - to discuss an offer of financial assistance for Mr Johnson.
That offer came from his wealthy Canadian friend Sam Blyth - also a distant cousin of Mr Johnson.
Mr Sharp - who worked as a government adviser at the time - has said he "simply connected people", and that his involvement in the loan facility ended at this point.
The warning to Mr Johnson was sent on 22 December 2020 - about two weeks before his government unveiled Mr Sharp as the new BBC chairman, the Sunday Times wrote.
This leaked Cabinet Office memo was reportedly sent after the two men asked for advice on Mr Johnson accepting an £800,000 loan from Mr Blyth.
It reportedly came from Mr Case, a top civil servant, around two weeks before Mr Sharp was announced in his BBC role on 6 January 2021.
Mr Johnson is said to have secured his loan the following month.
The memo allegedly requested that Mr Johnson "no longer ask [Mr Sharp's] advice about your personal financial matters", making specific reference to Mr Sharp's "imminent announcement" as BBC chairman.
The then-prime minister was reportedly told he could take out the loan - without declaring it - as long as his guarantor Mr Blyth had no "business or personal interests in the UK" beyond his family ties.
But Mr Blyth was on a government list of recommended candidates for the British Council, without senior figures in the public body realising his relation to Mr Johnson, the Sunday Times reported.