On Tuesday, deputy mayor Nobby Clark indicated that Mayor Sir Tim Shadbolt could be facing a vote of no-confidence.
The claim came hours after a closed meeting at which Sir Tim said he was bullied by councillors about a pending media statement.
The incident is the latest in a growing list of complications in Sir Tim’s ninth term as mayor.
He wonders "how much longer the (Local Government) minister can fiddle whilst Invercargill burns".
"For any local authority to function with any level of normality, you have to have understanding between the CEO, mayor and deputy mayor," he said.
"I cannot believe what’s happening."
Asquith believed the situation had reached a point worse than November last year, when Tauranga Mayor Tenby Powell resigned following the Tauranga City Council’s decision to bring in a Crown manager on the back of conflict and dysfunction within elected ranks.
Asquith said Invercargill needed similar action.
"What you really need is a clean sweep of everyone clearing out. You need a fresh start. When you have elections, you find the same people standing and the same people being elected."
Approached for comment, Clark called Asquith’s statements "outrageous".
Feedback from the Department of Internal Affairs said the council was now the "model" in New Zealand for resolving internal issues, he said.
"If you read the Thomson Report from six months ago, you will see this council has made some huge gains.
"And given that I was one of the people that was very frustrated at the council, things are working very, very well here."
The Thomson Report was a damning independent review of the Invercargill City Council released in October last year.
It highlighted a "leadership void" with Sir Tim at the helm, and strained relationships between the mayor, deputy and chief executive.
In Tuesday’s media statement, Sir Tim called the council a "regime" and accused an unnamed staff member of going through his personal documents held in storage at a council-owned building.
"All the councillors took turns to disgrace me for the use of this word [regime], which I consider appropriate," Sir Tim said.
Yesterday, the mayor said he stood by his comments about being bullied, and noted that no councillors or staff had checked on his welfare.
Council chief executive Clare Hadley denied the mayor’s claim that she appointed a staff member to undertake the task of sorting through his possessions.
Local Government Minister Nanaia Mahuta was approached for comment, but could not respond by deadline.
