Serviceteam is the preferred bidder for two contract areas, worth£250m, and Accord won the bids for two contract areas worth£125m.
Four unions - including the T&G - have accused the council of unfairly ruling out an in-house bid. The housing DLO has accumulated debts of£14m.
Leader of the city council Albert Bore (Lab) said: 'The preferred bidders are offering better efficiency and productivity. We need to move forward and deliver a better service to tenants.'