The Sheffield Star reports that the union has exrpressed 'grave concerns' over what it describes as preferential treatment given to CSL, the company which will be given the contract to run the service.
At a meeting yesterday, councillor Trefor Morgan put questions raised by workers and trade unionists, including asking how much the authority had spent on reviewing the service.
'Is it true that it exceeds£400,000 and that the projected saving through privatisation is only around£700,000?' he asked.
Councillor Jan Wilson said staff, who had been on strike for three weeks, were forcing the council's hand because it had a statutory responsiblity to keep the service running.
There was a call for the issue to be put to arbitration but the council's solicitor, Mark Webster, ruled that this was not possible.
Unison's demands for guarantees could not be given without illegally restricting councillors' future decisions.
Joan Barton, a Labour member, said her party had honestly tried to keep the service in-house by calling for a second review.
'Sadly while we believe we took an honest position, it was interpreted as a position of weakness, that we wouldn't decide until after the election,' she said. 'Well, we are making a decision now.'