The Tory party leadership has continuously sniped at Mr Dyke's campaign to secure the pounds 377,000 a year job because of his close connections to New Labour. Mr Dyke gave pounds 50,000 to Labour over five years.
Although Mr Hague has no official veto to any future appointment to the prestigious post, it would be difficult for governors to fill the post with a candidate so openly opposed by one of the main parties.
Mr Dyke has previously indicated that he is considering applying for the post but has rebutted allegations that he has discussed the matter formally with Chris Smith, the Culture Secretary. He wrote to The Independent to say that Mr Smith had given "slightly misleading" information in a Commons written reply about the two men's discussions over the prospective post.
The row over Mr Dyke has been part of the most controversial race for the post of the BBC's director-general in the corporation's history. Other candidates include Richard Eyre, chief executive of ITV and former group chief executive of Capital Radio, Alan Yentob, the BBC director of television and Tony Hall the chief executive of BBC News.
One senior Tory source said last night: "It would be unthinkable for William Hague, given the obvious cosiness between the BBC and Labour apparatchiks, to have the BBC headed by a notorious Labour apparatchik."