She took over as the chair of the BBC Trust last year.
Mrs Fairhead was a member of HSBC’s audit committee until 2010 and of its risk committee following that. She is currently chair of HSBC’s North American division.
Margaret Hodge, who chairs the PAC, raised questions about her BBC role last week.
Two weeks ago, HSBC’s chairman, Douglas Flint, appeared to blame Mr Meares for alleged collusion between the Swiss private banking division and its clients to evade tax.
Mr Flint told MPs on the Treasury Committee Mr Meares and Clive Bannister, who was boss of HSBC’s private banking operations until 2006, “certainly bear fairly direct responsibility for what went on in the private bank during their stewardship”.
At the same hearing, Mr Gulliver said the tax scandal had caused “damage to trust and confidence” in the company.
Mr Flint said he felt shame and would “take his share of responsibility” for the Swiss private bank’s failings.
The BBC’s business editor, Kamal Ahmed, says during the 2000s board members had little visibility of how the Swiss bank operated and that sources suggest they could not have been expected to have had.
He says there were strict rules in place protecting the confidentiality of Swiss accounts.
He also says it is the fact that the bank operated in such secrecy from its own board that will be of interest to MPs on the PAC and it is likely that this is the line of questioning the committee will follow.
Information about some 30,000 accounts at the Swiss private bank operation were leaked in 2007 to French tax authorities who passed it on to the UK tax authorities (HMRC).
HSBC has been involved in a range of banking scandals, including foreign exchange manipulation and rigging of international interest rate benchmarks.
When asked about the wider list of allegations and investigations into HSBC by international regulatory authorities last month Mr Flint said “it’s a terrible list”.
Despite reforms, he said he could not exclude the possibility of further problems emerging.
He said the task of reforming HSBC will “always be ongoing”.
Three senior HSBC bosses will face further questions from MPs on Monday over the tax scandal revelations at the bank’s Swiss private banking arm.
HSBC group chief executive Stuart Gulliver and the former head of the bank’s private banking division Chris Meares, will face MPs on the Public Accounts Committee (PAC) at 15:15.
BBC Trust boss Rona Fairhead will also face questions about her role at HSBC.
Mrs Fairhead has been a member of the bank’s board since 2004.