'The one thing we have all been clear about in England is that you could not conceivably establish elected regional assemblies as well as having a tier of shire counties and districts underneath,' Mr Straw said on Sunday.
Speaking on BBC1's On the Record, Mr Straw said this meant the timescale for the assemblies 'is longer than had previously been anticipated'.
Mr Straw, who is in charge of Labour's regional devolution plans, blamed the 'extraordinary inconsistency and incompetence' of the Local Government Commission for the delay.
'If it is not elected, which is the first stage, there is no problem in setting up a delegated body while the shires are disappearing,' the source said.
Labour had little new to offer voters if it hedged its commitment to inject democratic accountability into quangos.
This assessment was backed by Association of County Councils Labour group leader Ian Swithenbank, who has clashed with Mr Straw over his strong support for unitary districts and, in particular, one for his Blackburn constituency.
Mr Swithenbank said any future local government review was the responsibility of shadow environment secretary Frank Dobson and not part of Mr Straw's brief.
So far Mr Dobson has refused to commit the party to any new local government review before regional assemblies are introduced (LGC, 27 January).
Mr Swithenbank said Labour was still consulting on whether regions should be directly elected or made up of council nominees. 'I never know whether he is speaking for Blackburn or himself.'
However, in a recent interview with The Times, Mr Straw said businesses, as well as council representatives, would be invited to sit on interim assemblies.
These assemblies would oversee planning, economic development and transport, but he confirmed that directly elected bodies were unlikely in Labour's first Parliament.
Mr Straw's comments coincided with the publication of details of how much the civil service already spends and decides in English regions.
The information, from a series of Parliamentary questions by John Denham, Labour MP for Southampton Itchen, was collected to show the extent of unaccountable government already devolved to the provinces.
Mr Denham found that eight health service regions, 10 regional offices - integrating four other government departments - and eight regional offices of the Housing Corporation together spent more than £30 billion last year.
The government employs a total of 5,879 people in these organisations.