Two candidates will be on the ballot in the Apley Castle Ward of Hadley and Leegomery and five in the South Ward of Newport.
The vote will take place at the Leegate Centre for Leegomery, and at Newport Bowling Club in Granville Avenue.
The election will be held after councillors stepped down following the last election in May.
In Newport, one councillor will be on the ballot despite not wanting to be elected.
Andrew Griffin was standing in the election for Newport South as an independent councillor, but announced his intention to stop campaigning in the area.
But election law means he will appear on the ballot anyway, and could be elected if he receives enough votes.