Sir Kenneth, a former head of the Northern Ireland civil service, is understood to feel Mr Dyke's donation to the Labour Party will compromise the perception of BBC impartiality.
Baroness Young of Old Scone, the board vice-chairman, and Adrian White, a governor, released an unusualstatement claiming they were "entirely content with the painstaking and thorough process by which the board is selecting a new director- general. Neither of us have any intention of resigning."
All 12 governors have remained publicly silent during the appointment process and the vice-chairman's letter is unprecedented. Senior BBC sources believe it is a sign of the pressures being felt by the governors during the most bitterly fought leadership contest in the BBC's history. Several governors are unhappy at what they see as an attempt by the chairman Sir Christopher Bland, to bounce them into selecting Mr Dyke, chairman of Pearson television.
The BBC had to delay announcement of the new director-general until after the BBC's annual report next week. Now the announcement is unlikely until the first week of July.
Sources at the top of the BBC say the process is being extended by the chairman's efforts to get a majority behind Mr Dyke. Sir Christopher knows his position will be severely undermined if he does not get his man in place. "A huge row is underway," said a senior BBC insider. "At least two or three of the governors are deeply concerned about how Dyke would be able to represent the BBC in negotiations or battles with the Government when he is perceived to be so close to it. Many of them will resent being seen as Bland's pushovers. It is a very big struggle."
Mr Dyke's selection has also been made politically difficult, thanks to a campaign in some of the media to have him barred because he donated money to the Labour Party. William Hague, leader of the Conservative Party, wrote to the governors warning them he believed Mr Dyke's donations ruled him out.
Mr Dyke and Sir Christopher Bland are close friends who worked together at London Weekend Television in the Eighties where they both became millionaires.
The final round of interviews have been completed, but some may be called back if the governors seek a compromise.
Those vying for the job are known to include: Tony Hall, chief executive of BBC News; Alan Yentob, the corporation's director of television; Mark Byford, head of the BBC World Service and Richard Eyre, chief executive of ITV.Reuse content