The controversy stems in part from his continued membership of a Church of England group known as the Society, which does not recognise women priests.
The Archbishop of York, Dr John Sentamu, had previously said that the church would support his appointment.
Last week he received support from 36 members of female clergy in the Blackburn diocese, as well as three women bishops.
But local groups in Sheffield had protested his appointment. A new group called Sheffield Action on Ministry Equality wrote an open letter saying it had left them “deeply troubled”.