IT workers' union Unison is set to ballot all 5,000 of its members at the council to take action in support of the strikes.
The staff are resisting possible transfer to Capgemini or ITNet and want the option of being seconded to any new supplier.
The strike was described as a "PR disaster" last week by the National Outsourcing Association. "In Swansea they could be running the IT department on contractor fees for months, costing taxpayers thousands of pounds, and lending fuel to the anti-outsourcing argument," it added.
The potential escalation of the dispute was condemned by deputy council leader Gerald Clement, who accused the union of "holding the council and the people of Swansea to ransom".