If the government decides to give some of the BBC's licence fee money to other broadcasters, such as ITV and Channel 4, it is believed that Grade and Thompson, who have only been in their positions for less than a year, would resign.
The government's Green Paper, which sets out the future of the BBC, was supposed to be published tomorrow but has been delayed after a clash between former BBC director-general John Birt and media secretary Tessa Jowell about the corporation's funding.
Birt believes the licence fee money could be better spent by sharing it with ITV and Channel 4 to subsidise their own public service obligations.
His claims have put pressure on Jowell to include licence-fee sharing as a Green Paper option.
If the option is added to the paper it is likely that broadcasting union Bectu would strike because if the licence fee was cut there would be radical job cuts at the BBC, on top of the 6,000 made by Thompson last December.
The Green Paper will feed into the 2006 Royal Charter Review, which decides the BBC's funding for the next five years.
Earlier this month, Ofcom said that it did not think Channel 4 had an "immediate case for direct public funding" and told the broadcaster to develop cost-saving initiatives and expand its commercial ventures and alliances to meet its proposed £100m shortfall in funding.
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