Recycling collections are among a raft of council services under review for possible part-privatisation.
Building maintenance, security, janitorial services and event management – which all come under the facilities umbrella – are also under question.
But councillors will delay a decision until November 24, owing to concerns about the detail of the business case presented to them.
Council leader Jenny Dawe said: “Further clarification is required.”
John Stevenson, president of the Edinburgh branch of the trade union Unison, said: “We have been saying the process is dreadfully flawed, so any postponement of any part is to be welcomed.”