The framework, which is due to come into force at the start of 2015, is expected to be worth £500m-£1bn over four years.
LGSS is the shared services venture set up in October 2010 by founding partners Cambridgeshire County Council and Northamptonshire County Council, and ranks as one of the largest ventures of its kind.
Selected contractors will be in line for work from Norwich City Council and Northampton Borough Council, as well as Northamptonshire County Council and Cambridgeshire County Council.
Previous winner include Ashe, Borrass, Kier, Morgan Sindall, Galliford Try, R G Carter and Vinci.
This time around up to eight firms are expected to be named for each of five lots covering different project values.
LGSS construction framework
Band A: £100k up to £1.5m
Band B: £1.5m up to £3m
Band C: £3m up to £6m
Band D: £6m up to £10m
Band E: £10m and above
Full details of how to compete are available on the LGSS e-tendering system .
The deadline for submitting completed ITTs is 2 December. Further information can be obtained from the Due North helpdesk on: T: 01670 597 137