Figures obtained by the BBC showed that many councils faced a growing shortfall in their budgets.
The problem was most acute in county councils, which oversee the big spending departments of social care, children’s services and highways.
To try and close its budget gap, Norfolk County Council has earmarked £16.5m of savings from its adult social care budget.
Its children’s services department is facing cuts of £7m.
The authority is also looking at increasing council tax with three possible options, which range from hikes of 3%, up to 5% or more than 5%.
Mr Jamieson was asked by BBC Radio Norfolk breakfast presenter, Chris Goreham, for his response to Ms Mason Billig’s remarks that the council “hit 10 ages ago”.
He said: “I would not say that we have a lot of wriggle room, but what we do have is a clear plan for making these pressures as clear as possible.
“But what we need the government to do in order to help us do that… is a multi-year settlement.”
In a statement issued after the interview, Mr Jamieson said the council was “holding our own” at the moment.
But he added: “If I were playing penny falls in the arcade at Hunstanton, we would not be one of the 16 councils that have stated publicly that they would go over the edge in the next 12 months.
“However, there is no doubt we are on the board.”