His appointment comes amid a debate about the BBC licence fee and how the broadcaster is facing competition from streaming services.
Mr Sharp, who was once UK chancellor Rishi Sunak’s boss, will take over from Sir David, who will stand down in February, according to BBC media editor Amol Rajan.
The new chairman will work closely with new director-general Tim Davie, who is the former chief executive of the BBC’s commercial arm, BBC Studios.
He took over from Lord Tony Hall in September and said the corporation needs to keep reforming “with urgency” and stressed it must be “a universal public service”.
The broadcaster currently faces scrutiny over equal pay, diversity, free TV licences for the over-75s and competition from streaming services such as Netflix, as well as the ongoing coronavirus crisis.
Julian Knight, chair of the Digital, Culture, Media and Sport Committee (DCMS), commented on the reports of Sharp’s appointment.
He said: “It is disappointing to see this news about the next BBC chairman has leaked out ahead of a formal announcement from the Department of Digital, Culture, Media and Sport.
“The committee previously expressed some concerns over the appointments process, calling for it to be fair and transparent.
“The DCMS committee looks forward to questioning the preferred candidate for the post in a pre-appointment hearing next week on their views at a critical time for the BBC about its role and the future of public service broadcasting more generally.”
Culture secretary Oliver Dowden told Sky News’s Sophy Ridge On Sunday late last year that the Government wanted a “strong, big person who can hold the BBC to account” to be the broadcaster’s next chairman.
Mr Sharp has not yet commented on the appointment, which would see him earn a salary of £160,000 for “three/four days a week”.
According to the job description posted online last year, the BBC board “meets at least 11 times a year”. – PA