Libraries across Southampton in Hampshire closed at 12 noon on Wednesday of last week as workers struck against cuts.
This was the fourth strike held and was supported by 100 percent of Unison union members.
The union represents 90 percent of library staff.
Around 100 strikers and their supporters lobbied councillors as they arrived for a council meeting.
A Unison delegation addressed councillors and handed in a 4,500-strong petition opposing the closure of Millbrook and Thornhill libraries and the introduction of volunteers to replace paid library staff.
Councillors then debated a motion moved by Labour councillor Derek Burke supporting the dispute.
Although the motion was supported by Labour and Liberal Democrat councillors, it was voted down by Conservative councillors.
Within minutes of the debate, a senior manager from the council had contacted the Acas conciliation service, agreeing to talks aimed at resolving the dispute.