Some council leaders are keen to offer staff a five per cent pay rise to avoid strike action that could see waste pile up uncollected.
Some of the biggest local authorities believe the offer should be stepped up but that the Scottish Government should provide more funding for it.
City of Edinburgh Council's leader Cammy Day told BBC Scotland that the government should "dig deep".
Meanwhile, Glasgow SNP leader Susan Aitken said a higher offer should already be achievable.
A fresh offer was made by local authority body Cosla on Friday afternoon, after an initial 2% pay increase was rejected.
Its new proposal included an offer equivalent to a 3.5% increase.
GMB warned that industrial action to be taken by Edinburgh waste workers from August 18.
Other councils are set to join the strike days later from August 26 to 29 and again from September 7 to 10.
"Councils have been cash-strapped for many years. We are cutting services to the bone if we make any more cuts here in Edinburgh.
"We are calling on the Scottish government, who we know have hundreds of millions of pounds in reserves, to dig deep.
"If ever there was a rainy day that rainy day is now."
Ms Aitken said she was "disappointed" by the offer.
She added: "I don't think it is enough.
"I don't think it is a credible or realistic position for us to take to our staff and I think it is going to cause more delay.
"I would much rather have seen a bigger offer to be put on the table, but Cosla has made its decision."
GMB members will strike in Aberdeen, Angus, Dundee, East Ayrshire, East Lothian, Falkirk, Glasgow, Inverclyde, Highland, Midlothian, Orkney, South Ayrshire, South Lanarkshire, West Lothian, Perth and Kinross, and North Lanarkshire.
Unison members in Aberdeenshire, Clackmannanshire, East Renfrewshire, Glasgow, Inverclyde, North Lanarkshire, Stirling and South Lanarkshire have also said they will walk out.
Scotland’s Social Justice and Local Government Secretary Shona Robison said the latest proposal is “extremely disappointing”.
She said: “Despite the significant additional resources we have made available – more than half the amount Cosla asked us for in order to make a 5% offer – we understand there is only a 3.5% offer on the table.
“We urge Cosla to urgently reconsider its position to avoid industrial action.”
Ms Robison added: “Despite the fact the Scottish Government’s budget for this year has been cut by the UK Government, we’ve allocated more money to local government.
“And last week an extra £140 million was committed on a recurring basis to support a higher pay award for council staff.
“While I understand the challenges that local authorities face, the Scottish Government must balance a fixed budget with very significant competing demands as a consequence of the cost-of-living crisis and the inaction of the UK Government.
“As well as a fixed budget largely determined by the decisions of the UK Government, our main tax levers are set for the whole year and cannot be changed.
“We have no power to borrow for this spend. So this extra £140 million has got to come from somewhere else within our budget and no more funding can be offered.”
GMB senior organiser, Keir Greenaway, said: "For our members, all they care about is getting a pay rise that will get them through this cost of living crisis.
"At the end of the day, they are more worried about paying their next bills than which government department the money comes from."