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Sharp has recently been acting as an adviser to finance minister Rishi Sunak, Sky said, adding that he was likely to take up his new role in mid-February.
He will succeed David Clementi, a former deputy governor of the Bank of England, who is stepping down in February after four years as chairman of the broadcaster.
Queen Elizabeth officially appoints the BBC’s chairman on the recommendation of the government.
Clementi recruited the BBC’s new director general Tim Davie, who started in the role in September.