The project is being brought forward by the city council at a site in Princess Road, Westbourne.
Kier will build 121 flats and a hostel across four blocks rising to up to 10 storeys.
Around 86 flats would be made available by the council at ‘affordable’ rates. The remainder would be provided at normal market levels.
Alongside the flats, 20 units of short-term accommodation are also to be built to temporarily house families at risk of becoming homeless until more permanent alternatives are found.
Kier is expected to start work in the Spring.