The Yorkshire authority tendered for an innovative recycling company to join it in a public-private partnership to improve the way it deals with its waste.
The citys current system of separate contracts for individual waste streams has been branded ineffective by the council.
It has received lots of applications from firms wishing to improve on this, and is willing to listen to all sorts of proposals.
Principle procurement manager Trevor Black said: We have received a large number of responses and do not want to pre-judge anything lets see whats out there.
At the moment we have lots of different arrangements and its not really operating as successfully as we would want.
It is going to be a long procedure to look through all the applications and judge their different costs and risks. We have no idea how long it could take.