Gavyn Davies was the first to resign after the report
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The government is advertising for a new chair of the BBC governors in three Sunday newspapers.
It follows the departure of Gavyn Davies in the wake of the Hutton report which severely criticised the BBC.
Lord Hutton's inquiry into the events around the death of Dr David Kelly provoked fears the corporation's independence could be threatened.
Applications must be in before the end of the month, and a decision is expected by mid-April.
Critics have questioned the dual role of the governors as both regulators of the corporation and defenders of its independence.
The advertisement warns the successful candidate will lead the BBC governors at a difficult and challenging time.
He or she will have to oversee the BBC's responses to the Hutton report and the review of its royal charter and, with the board of governors, choose a new director general to replace Greg Dyke.
A shortlist of candidates for the chairman's job will be interviewed by a panel drawn up by the Department of Culture, Media and Sport, who will then recommend a choice to ministers.
Following the Hutton fall-out, the process is also being scrutinised by a second panel drawn up by the commissioner for public appointments.
Culture Secretary Tessa Jowell says it would not be right to choose someone actively political but she said she is reluctant to rule out those with past political affiliations.
Lord Hutton found the Today programme report by BBC journalist Andrew Gilligan which claimed the government had "sexed-up" its Iraq dossier was "unfounded".
It also found that BBC management had failed to properly investigate whether that report was accurate and fair.
This quickly led to the resignations of Mr Davies, Mr Dyke and Mr Gilligan.
But acting BBC Director General Mark Byford has insisted the BBC's independence has not been threatened by the fallout from the Hutton Report.
He told BBC's Radio 4 Today programme the fact the corporation had admitted making mistakes would only make it stronger.
He said the BBC had faced pressure for decades but the key was "to be robust and stand up to it".