Bosses and a union at Derby City Council have disagreed over how effective industrial action has been.
Staff are unhappy that a car allowance scheme is to be replaced with a mileage rate.
Managers insisted fewer than 100 people have refused to use their vehicles for work, less than 5% of those who are affected, since action began on 9 June.
But Unison said the figures were unreliable and more than half of the 800 essential users had taken action.
The dispute centres on the council's move to stop the allowance, which totals more than £1,000 a year for some staff, and replace it with a flat fee of 40p-a-mile.
Unison, which insisted the allowance was an essential part of the salary package, began industrial action last week.
But the council has said fewer than 100 of about 2,000 of its essential and casual car users had taken part and contingency plans were working well.
Council leader, Harvey Jennings, added: "It is clear that the action is only being supported by a small number of employees.
"These employees are using this opportunity to express their strength of feeling at the changes, which is their right and we respect that. However, these changes are necessary in the current climate and will need to be made on 1 July."
But Charlie Carruth, regional organiser for Unison, said: "I don't know how they got those figures because they haven't done the work to stand them up.
"While the action is better supported in some areas than others, we believe more than half of the council's 800 essential users, those who need their cars more than twice a week, are taking part."