Culture secretary Jeremy Hunt has personally interviewed the candidates and is reported to have submitted a formal recommendation to David Cameron.
The Department for Culture, Media and Sport has refused to comment on reports that Lord Patten (pictured), a former Conservative party chairman under John Major and ex-governor of Hong Kong, is to get the job.
Patten has been chancellor of Oxford University since 2003. Other candidates tipped for the job included former Financial Times editor Sir Richard Lambert and ex-Radio Authority chairman Richard Hooper.
The Department for Culture, Media and Sport said in a statement: "The appointment process for the chair of the BBC Trust is ongoing. We will not comment on speculation at this stage.
"We expect to be in a position to announce the preferred candidate in late February or early March."
The successful applicant will be grilled by a cross-party group of MPs on the Commons culture, media and sport select committee, chaired by Conservative MP John Whittingdale, on March 10.
Sir Michael Lyons' term as Trust chairman ends in May. The former chief executive of Birmingham city council took up the job in 2007 following Michael Grade's sudden move to ITV.
Hunt has suggested that future appointments to the BBC Trust should be a parliamentary decision and not just a ministerial one.
He told a Voice of the Listener and Viewer conference in parliament earlier this month: "I'm a very strong believer in parliament having more involvement in the selection process for the person who runs the BBC Trust."
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