Bob Wellington said budget cuts in the past five years had been "horrendous".
He reserved his criticism for the UK government, saying the Welsh Government was "restrained" in what it could give councils.
The Labour councillor will stand down as Torfaen council leader next month.
Mr Wellington is Wales' longest serving council leader, having served for 13 years. When he leave his post, he will also stand down as leader of the Welsh Local Government Association.
Councils are set to get their first cash increase since 2013-14 in the funding they get from the Welsh Government next March.
However, once inflation is taken into account, the overall Â£4.1bn funding pot amounts to a real-term cut compared to the year before.
The Welsh Government gets most of its money for councils from the UK government through a block grant.
Asked what he thought the state of local government would look like in 10 years, Mr Wellington, who represents the Greenmeadow ward in Cwmbran, said: "The way we are going there won't be such a thing."
"There needs to be a change of heart on a national level, and if there isn't there will be local authorities in Wales that will go to the wall," he said.
"If we continue to have these cuts on a continual basis there will be serious problems stacking up in Wales," he said.
"I lay the blame on the national [UK] government with an austerity programme that's being passed down to the assembly.
"They are having difficulty and they are obviously restrained in terms of what they can give local government."
Mr Wellington added: "The last five years have been horrendous, not only for me but for all leaders of all local authorities in Wales."
Last month, Finance and Local Government Secretary Mark Drakeford said no council would "have to manage on less than 99.5% of the cash provided to them last year".
"When added to the other sources of income available to them, many councils will be able to increase their spending next year," he said.
Councils in Wales will "go to the wall" if there is no change of heart on austerity, the outgoing leader of the Welsh Local Government Association has said.