The ruling Lib Dem/SNP administration is set to overturn the recommendations from officials, who proposed 875,000 worth of savings, because they were "simply never an option".
However, the Evening News understands that at least seven organisations, including two which offer services for asylum seekers and people with HIV or Aids, still face losing a huge chunk of their grant money.
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Hide AdOthers, including the closure-threatened Gorgie and Dalry Partnership, remain uncertain of their future.
The full list of cuts will be announced later this week.
Opposition politicians today said the uncertainty created over the past few weeks had destroyed the voluntary sector's trust in the city council.
But administration councillors accused the Labour group of "scaremongering".
The threat of funding cuts was first revealed in the Evening News, which obtained a copy of a leaked report drawn up by council officials.
Councillor Gordon Mackenzie, the city's finance leader, said today: "Some of the proposed cuts reported in the media were simply never an option for this administration."
Deputy council leader Steve Cardownie added: "We set out to try to protect the voluntary sector. It's not been possible in every case, but we believe we have managed to limit the damage.
"It's unfortunate that Labour deployed the tactic of spreading fear that major (cuts] would be made to grants – that's proven not to be the case."
The Ca(i)re Project at the Eric Liddel Centre in Morningside Road, which offers support, training and educational courses for hundreds of carers, is one body set to avoid a cut, which could have totalled 27,064.
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Hide AdCo-ordinator Sunil Bhatnagar, said it was "fantastic news".
"People were very concerned because the project was facing closure," he said.
We would prefer to hear about (the reprieve] from the council, but it's brilliant."
One Parent Families Scotland is another organisation set to avoid a cut – of 1316.
Director Sue Robertson said: "We are delighted the council won't reduce the grant. It's a relatively small amount of money, but it's very pleasing because funds are extremely tight."
Full details of the council grant cuts remain unclear, although the administration has put its total savings at around 780,000.
The Evening News understands that Bingham and District Older Peoples Project may lose half its funding.
It is thought the Hutchison Vale Sports Club, which provides sport and leisure activities for young people, is also to lose around three-quarters of its 1150 grant.
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Hide AdLes Trotter, an official with Hutchison Vale FC, said: "The club doesn't have a lot of money, and is run by volunteers. We are always looking for money for equipment and kids' strips."
Labour group leader Ewan Aitken warned these cuts were just "the thin end of the wedge".
He said: "The council should have talked to groups to understand the impact of this, and they haven't done that."
He added: "We spent years building up trust with the voluntary sector, and it has gone in the nine months since the election."
WINNERS
Ca(i)re Project
Cavalry Park Sports Club
Living Memory Association
One Parent Families Scotland
Sports Awards
North Merchiston Club
Voluntary Organisations Training
Score Scotland
Edinburgh International Science Festival
Gorgie City Farm
Royal Zoological Society
Workers' Education Association
West Edinburgh Action
Edinburgh World Heritage Trust
Nisus Scotland
Edinburgh and Lothians Greenspace Trust
Water of Leith Conservation Trust
Women onto Work
Edinburgh Cyrenians
LOSERS
Positive Voice
Calder Community Centre
Calton Youth Ministry
Hutchison Vale Club
Tynecastle Club
National performing companies
UNCERTAIN
Bingham and District Older People's Project
Gorgie and Dalry Partnership