The deals continue a long-running association between Southampton City Council and the firms, which have been delivering upgrades in the city since 2006.
Connaught was also one of the other preferred Decent Homes contractors until in went bust.
Seven other firms bid the deal which will see Drew Smith cover homes in the west subcontracting out a third of the work and Mitie upgrade homes in the east of the city with just 5% expected to be subcontracted.
The proposed programme of refurbishment will include but not be limited to kitchen and bathroom upgrades which will incorporate associated electrical wiring, replacement heating, asbestos removals and surveys, and other general building works.
In order to standardise the kitchen units installed across its estate, the council will specify the supplier of kitchen units.