Mr Swinney is understood to have won over sceptical council leaders after offering them the extra money, alongside a promise of greater autonomy over how they spend the cash.
Sources close to the talks have revealed he told councils they will receive a rise of 400 million on this year's budget of 10.7 billion. That will be enough to bring in the freeze, and so fulfil a key SNP manifesto pledge.
Although Mr Swinney's offer was accompanied by a threat to give them only half the rise if they refused to sign up to the freeze, The Scotsman understands council leaders will recommend the deal to Scotland's 32 local authorities.
On top of the 400 million, the minister told local authorities they could keep cash previously taken out of their budgets as an assumed efficiency saving. The sum was 168 million this year but is likely to be more next year.
In a further concession to councils, Mr Swinney is understood to be prepared to end what is known as the "ring-fencing" of central-government funds.
About 1.5 billion a year is given to authorities on condition that they spend it in certain areas - for example, helping disadvantaged children's education or on school buildings.
Ministers are prepared to include this money in councils' overall budgets, allowing them to decide their own priorities - something local authorities have long campaigned for.
Last night, both the Scottish Government and councils were refusing to go into details of their meeting in Edinburgh.
A spokesman for Mr Swinney said there could be further talks ahead of a formal announcement on council funding in the Scottish Government's budget next Wednesday.
Leaders of the Convention of Scottish Local Authorities (COSLA) are understood to be delighted that the deal will give councils the first rise in the proportion of total Scottish Government spending for many years.
At present, they receive 33 per cent of the total budget, and the rise of nearly 4 per cent will be higher than for other policy areas, such as transport or the environment.
Senior council leaders will meet formally on Monday to discuss the deal, before it goes to a meeting of the leaders of all 32 authorities on Friday.
Although COSLA cannot force councils to freeze taxes, its expected endorsement of the policy will increase the pressure on them to bring in the freeze.
The deal is a major coup for Mr Swinney, who has put a massive effort into securing the agreement - including persuading sceptical councillors in his own party - which is vital to the SNP's credibility as a government.
The SNP has made great play of the fact the council tax went up by 60 per cent under the previous Labour-Liberal Democrat Scottish Executive.
The agreement also allows the Nationalists to boast about implementing a freeze in the unpopular tax, at a time when they are struggling to bring in other key manifesto pledges.
At Holyrood yesterday, the row over the SNP's manifesto promise to fund 1,000 new police officers continued.
Scottish Government insiders revealed that Kenny MacAskill, the justice secretary, appeared to have won the finances to fund the recruitment of only just over 200 additional police officers next year.
In order to protect the funding of the services, the cash which funds police authorities but is paid via councils will continue to be ring-fenced.
Nicola Sturgeon, the deputy First Minister, insisted yesterday that the SNP would fulfil its manifesto pledges.
She said the Nationalists had kept their commitments to reverse planned accident and emergency unit closures, abolish the graduate endowment and scrap tolls on the Tay and Forth road bridges. Standing in for Alex Salmond at First Minister's Questions because he was in Sri Lanka, Ms Sturgeon told MSPs: "The pledge we made to freeze council tax will also be delivered.
She went on: "In just six days' time, when we publish our budget, then all the waiting will be over and all the other parties in this chamber will see exactly how we intend to honour that commitment."
Annabel Goldie, the Scottish Tory leader, challenged Ms Sturgeon over a promise the SNP made in opposition to freeze the tax.
Ms Goldie claimed that, now the Nationalists were in power, they had "ducked and dived, bobbed and weaved and they've broken their pledges one after another". But the Deputy First Minister told her: "Not only did I give that guarantee a year ago, I'm going to say the same thing today - this government is determined to freeze council tax."
She added: "Under the last government, council tax went up by 60 per cent. I think the people of Scotland have had enough of council tax rises under Labour and the Liberals. That's why they want to see this government freeze council tax."
Freeze vital to show party can deliver what people want
FREEZING the council tax was a flagship pledge in the Scottish National Party's manifesto for the Holyrood elections.
The SNP's private polling told the party the tax was deeply unpopular, particularly among older people. The Nationalists promised to abolish it and replace it with a local income tax.
However, party strategists realised this could not be achieved immediately and the policy of freezing the council tax was agreed as an interim measure.
In opposition John Swinney said he would impose the freeze using laws introduced by the Tories to deal with high-spending councils.
However, in government the finance secretary quickly backed off from that threat. Instead he sought to persuade local authorities to help him implement the commitment.
To demonstrate how important this was, one of the first meetings Mr Swinney had as a minister was with COSLA. And he has gone out of his way to charm local authority leaders, visiting more than half of Scotland's councils during the summer.
Mr Swinney has emphasised he believes councils are equal partners with central government in the governance of Scotland.
This was music to the ears of councillors who believe the previous Executive had treated them as second-class citizens, despite holding their own electoral mandate.
As well as offering councils more money, Mr Swinney has promised to give them more leeway on how they spend it and hand them powers that currently rest with quangos.
With the SNP struggling to deliver its promises on extra police and class sizes, securing a deal on the council tax freeze is vital to the new administration anxious that it can use its position as a government to introduce popular, not to say populist, policies.