The calls to promote Mr Cook are motivated by a belief that Mr Blair needs to sharpen Labour's Shadow Cabinet. The shadow Trade and Industry Secretary - who has embarrassed the Government over the Archer affair - is highly rated as an adversary by the Tories.
Mr Blair is determined to resist the pressure and keep Mr Brown in place. They have agreed an economic strategy, which could be destabilised by a change of shadow Chancellor, and they are close friends. Mr Brown stood down to allow Mr Blair a free run in the Labour leadership contest.
The pressure for Mr Cook to be promoted is coming from within the ruling National Executive Committee, including the trade unions, but the possibility of Mr Brown being moved sideways is also being discussed privately among Shadow Cabinet colleagues. Mr Cook may be given the Home Affairs portfolio to attack the Tories on law and order, on which Mr Blair outshone Michael Howard, the Home Secretary.
Mr Blair will shuffle the team after the Shadow Cabinet elections in mid-October. His supporters say there will be widespread changes to impose his stamp on Labour's front bench appeal. Dawn Primarolo and Ann Clywd may win seats in the Shadow Cabinet, and Jack Cunningham, the shadow Foreign Secretary, is vulnerable.
The tussle over the shadow Chancellor's portfolio goes to the heart of Labour's strategy for the next general election. There has been a long-running campaign against Mr Brown by the left of the party, who believe he is too cautious. They want more radical plans to tax the rich and spend more on measures to create employment and alleviate poverty.
Mr Brown has laid out his own principles for taxation, with a commitment to 'fairness'. Some of his senior colleagues want to use those words to tie Labour to an election commitment to higher taxes on those who can afford to pay more.
Backing Mr Brown, the Blair camp feels that to match the tax strategy laid out this week by the Liberal Democrats, hitting the middle classes, would be suicidal for Labour, particularly after opinion polls showing Mr Blair's leadership is winning support in the South.