Dimbleby, who has worked for the BBC for 40 years, is understood to be one of about five shortlisted candidates to succeed the current chairman Sir Christopher Bland.
The shortlist is also believed to include the BBC deputy chairman Gavyn Davies, Baroness Jay and the former head of Channel 4 Michael Grade.
Earlier this year, Dimbleby sold his family's newspaper group, the Dimbleby Newspaper Group, to Newsquest, the UK regional newspaper company owned by US publisher Gannett.
The Dimbleby family owned the newspaper group for more than 100 years. The group published nine titles in upmarket South West London. The titles included the Richmond & Twickenham Times and the Teddington & Hampton Times.
Dimbleby, who currently presents the political programme Question Time, could have an advantage over some candidates because he has always guarded the secrecy of his political leanings.
Greg Dyke, the BBC director general, is known to have strong Labour sympathies and the appointment of another Labour sympathiser, such as Baroness Jay or Gavyn Davies, could raise questions about the BBC's impartiality.
Sir Christopher steps down in September to become chairman of BT. His successor will be chosen by a panel of civil servants at the Department of Culture, Media and Sport.
The selection has been put back because usually the leader of the opposition is involved. The final selection must be approved by the Prime Minister.