Falconer is the favourite to take up the role because of his political clout, but Blackstone is thought to be interested. CABE is known to have discussed the post with both ministers.
A source close to the project said: "The ministers will have to sit down and decide between them, with the advice of their civil servants, who should become the champion of champions."
He added that the choice would have to come from Falconer's and Blackstone's departments, as they are most closely involved with building design.
Blackstone, also know as Baroness Blackstone of Stoke Newington, served in Harold Wilson's government and sat for 10 years on the board of the Royal Opera House.
The news comes as the appointment of departmental champions nears completion.
The only departments still to select someone are Northern Ireland, Work and Pensions, and International Development, where there are only two ministers.
Departmental champions include Lord Sainsbury at the DTI, Lewis Moonie at the Ministry of Defence, Jane Kennedy at the Lord Chancellor's Department, Yvette Cooper at the Department of Health and Denis MacShane at the Foreign Office.