The Conservative administration and Labour group have voted to refer the consultation to their constitutional advisory panel as they believe it is flawed.
The largest sample, a phone poll of 500 residents, showed 41% favoured a mayor and council manager, 33% a mayor and cabinet and 21% a cabinet and leader.
Other parts of the consultation, however, received an exceptionally low response. Only 85 people responded to a questionnaire sent to 33,000 homes. Of these, 35% favoured mayor and manager, 46% mayor and cabinet, and 20% cabinet and leader.
are a number of unusual things about
the consultation. Only 6% know anything about the changes - normally it's between 10% and 20%.
'One of the questions was whether the council should be led by a group of people rather than one or two. Nearly nine-tenths said it should be led by a group of people and 77% said elected councillors should be involved in the election of the person that leads the council.'
It was too early to say whether or not the council would reject a mayoral referendum and stick with the cabinet-and-leader model, but this was an option, he added.
Lib Dem group leader Linda Cufley said under the current cabinet-and-leader system 'it is very difficult for the public to know what's going on. I think there might be some sort of backlash against that'.
Society of Local Authority Chief Executives president James Hehir said: '[The mayor-manager option] is an opportunity for two people together to really drive the council and that's good. But on the other side it would be more difficult to get full council support for initiatives.'
Shepway must submit its proposal by 14 September.
THE THREE MODELS
Leader and cabinet: a leader elected by members and a decision-making cabinet of up to 12 members with individual portfolios.
Mayor and cabinet: a directly elected mayor and cabinet of members - the two must work together.
Mayor and council manager: a mayor supported by a single manager with direct responsibility to carry out his or her plans.