Council leaders have submitted a new pay offer to unions following lengthy negotiations aimed at averting further strikes around Scotland.
The dispute has seen piles of rubbish build up in city centres as waste workers went on strike.
Talks between the First Minister, unions and local authority bosses went on until late on Thursday night at St Andrews House in Edinburgh.
Leaders from Cosla began talks at 9am on Friday to discuss a new offer to be put to the unions.
That fresh offer, the full details of which have not yet been published, was received by unions in the afternoon.
Strikes due to go ahead next week could be suspended if the offer is accepted by union representatives.
Cosla’s resources spokeswoman Katie Hagmann said: “Firstly I would like to thank all our trade union colleagues for the constructive discussions.
“The revised offer made shows that Scotland’s council leaders have listened to the concerns of our workforce and have responded positively.
“Council leaders have said consistently throughout these negotiations that we very much value and are grateful to the local government workforce.
“We have sent letters to our union colleagues following today’s meeting and hope that this enables strike action to be suspended and allows our workforce to get back to doing what they do best, delivering high-quality essential services for the people within our communities right across Scotland.”
Labour MSP Mark Griffin said of the revised deal: “This news will come as a relief to millions of Scots and our hard-pressed council workers – but the blame for these weeks of chaos lies squarely with this SNP Government.
“It is simply not good enough that it took weeks of industrial action to embarrass the SNP Government into action.
“This will not be the last industrial dispute of the year. With sector after sector balloting for action, the SNP Government must realise that it cannot continue to approach industrial relations in this slapdash and damaging fashion.”
Earlier on Friday, a trade union official said the talks at St Andrews House had been “constructive” despite ending without a deal.
Johanna Baxter, head of local government for Unison Scotland, told BBC Radio Scotland’s Good Morning Scotland programme that things were more optimistic than they were 24 hours prior, and there was a “willingness from all sides to make movement”.
She added: “I think the judgment for Cosla leaders this morning is whether they have enough to deliver what we need for our members.”
She said the marathon talks on Thursday, which were chaired by Nicola Sturgeon, looked into how much money was on the table and how it was constructed.
“The difficulty obviously has been the money, and how much money has been on the table.
“The difficulty we face particularly in the last week is the fact that however you cut the 5% that was on the table in terms of consolidated money, and that is recurring, then you couldn’t get a flat rate that worked for the majority.”
A spokesman for GMB Scotland said: “We believe progress has been made and would hope last night’s negotiations will result in a fresh offer that can be put to our local government committee for review.”
Further strikes by waste workers are planned for next week, with staff at a number of local authorities expected to walk out for eight days.
Workers in schools and early years learning will also take part in a three-day strike in some areas.