The government has indicated it would agree to postpone elections due in May until 2027, if authorities make a request by mid-January.
One of the councils that has taken up that offer is Nuneaton and Bedworth Borough Council, which would also be affected by the Warwickshire changes.
There, Labour group leader Chris Watkins said: “It was felt that holding [them] for a local authority that is shortly to be abolished is not a prudent use of resources.”
The Labour group in Rugby said the limited life-span of Warwick Borough Council and the challenges of implementing the election were the main reasons for requesting the delay and noted the requests to delay elections elsewhere.
Councillor Maggie O’Rourke, of the Labour group, also said she was concerned the council would be hit with staff cuts as the re-organisation approached.
But Derek Poole, Conservative leader of the opposition on the council, said the re-organisation was “not relevant” and told the Labour group it was “down to the voters” to choose whether the elections continued, “not you”.