George Osborne is the latest name to be linked with the job of BBC chairman, according to reports.
The former chancellor and Conservative MP is being encouraged to apply for the role, the Daily Telegraph reported.
Osborne, who stepped down as editor of the Evening Standard in June after three years, is not believed to have been approached over the vacancy yet.
Ministers increased the salary this week for the part-time position to £160,000 a year. A source told the newspaper that ministers could further increase the pay packet to £280,000 in order to tempt Osborne.
The BBC declined to comment on the latest speculation.
David Clementi, the current chairman, is due to step down when his term in office ends in February, after overseeing the appointment of Tim Davie as the new director general.
Other names linked to the role include David Dimbleby, the former Question Time host, Trevor Phillips, the former chair of the equalities watchdog, Lady Morgan, the former culture secretary, and Robbie Gibb, a former head of the Downing Street press office.
It comes after Charles Moore, the former Daily Telegraph editor, ruled himself out earlier this month.
Moore, whose writings on race and Islam have come under renewed scrutiny since he was publicly associated with the role, is understood to have backed away amid family health problems despite claims he was Downing Street’s top choice for the position.
Reports had said that Moore, who was once fined for refusing to pay his television licence fee in protest at the BBC’s content, would have wanted substantially more than the current £100,000 licence fee-funded salary.
Government sources have consistently emphasised that the formal recruitment process for the position has yet to begin.
The speculation about Moore raised concern within the BBC about the direction that he would seek to take the corporation. Julian Knight, the Conservative MP who chairs the culture select committee, said appointing a man who refused to pay a licence fee as BBC chairman was “like being convicted of fraud and being in a bank”.