Coun Driver is one of four people who were re-bailed by police in August in connection with the police inquiry into the council’s contracts with the organisation One Connect.
He is still serving as leader of the county council.
Another arrested in the same probe – Ged Fitzgerald, the current Liverpool City Council chief executive and former Lancashire County Council chief executive – has been suspended.
Now the Lancashire Association of Trades Union Councils has lodged a Corporate Complaint with the county council.
In a letter, the body said: “We consider that following the decision of Liverpool City Council on the 18 September 2017 to suspend Ged Fitzgerald... Lancashire County Council should review whether it is acceptable for Councillor Driver to continue to hold the position of Leader at this time.”
One Connect was a partnership between Lancashire County Council and BT which was set up under the previous Conservative administration.
All the men deny any wrongdoing.
The others arrested were former Lancashire County Council Chief Executive Phil Halsall, and David McElhinney, who was chief executive of One Connect.
A spokesman for the county council said the complaint did not fall within the terms of the Corporate Complaints Procedure, which relates to services provided by LCC, therefore no further action would be taken.
Coun Driver told the Post today the two cases could not be compared in this situation.
He said: “The difference is I am an elected member. There’s no comparison.”