"Worse still, they insist that at my age I'm unfit to serve in office."
He went on to challenge the other councillors aged over 70 to a 100m sprint, a 50m swim and a lap of the ILT Stadium velodrome.
"Hopefully this will put an end to all these allegations."
In the column, he said it was his outspoken nature which was upsetting those against him.
"I believe the real reason they want to remove me from office is because my critics want to remove people like Donald Trump, Boris Johnson and Winston Peters and myself is because we are prepared to speak out and tell the truth about what is going on in their respective countries."
Sir Tim, as well as the ICC chief executive and fellow councillors, have come under close scrutiny lately.
The Department of Internal Affairs (DIA) recently intervened after media articles and correspondence received by the department and Minister of Local Government highlighted "significant conflict" among elected representatives and the chief executive.
The council is at present being evaluated by an independent reviewer in line with the DIA request.
Sir Tim is also suing the council for $448,000 in relation to the defamation case former Invercargill city councillor Karen Arnold brought against him in 2015.
In July, he defended his effectiveness as mayor, saying he was "fighting fit".