The council announced an independent review found school meal prices would need to increase by more than one third in order for the in-house school meals service to break even.
Schools and academy trusts are free to select their own supplier. The local authority hopes the external provider will take on the staff currently employed in local schools.
Sarah Hill, cabinet member for finance, anti-poverty and corporate services, said: "City Catering has served our schools well for many years, but rising costs and stronger competition mean it is no longer possible to run the service without significant subsidies.
"We will support schools and staff throughout this process to make sure the transition is safe, steady and focused on what matters most - good meals for children."