At the Local Authority Tenders Bill's third reading in the house on Friday, Tory whip Stephen Day confused it with Lord Hunt's Local Authority (Experimental Arrangements) Bill and objected, despite the Tories earlier agreeing not to.
Ms King had lobbied hard for Tory support after backbencher Eric Forth blocked the Bill in the report stage.
The Bill would give councils the ability to stipulate staff-friendly policies in awarding contracts, and has the support of councils, unions and contractors. It will get another reading on 3 July, but this will leave little time for it to clear the Lords, Ms King said.
Under the Bill, the environment secretary could give councils power to consider non-commercial issues in letting contracts. The government has agreed contract compliance rules need to change but primary legislation would take two years to take effect.
Local government minister Hilary Armstrong has said the government would agree to contract considerations that have the backing of all parties.
Business, councils and unions have agreed that quality, equality and workforce issues should be considered. Joint working groups will meet on Tuesday to try to agree detail on pay and conditions, health and safety, quality and training, and equalities.