Council chiefs are demanding talks with the Deputy First Minister on the key issue of teacher numbers.
David O'Neill, the president of the local government body Cosla, hit back after Mr Swinney told local authority leaders he will be forced to "claw back" funds from them if they fail to protect teacher numbers.
Councillor O'Neill said council bosses have been left angered by the "threat of financial sanctions" from the Deputy First Minister.
He said they are "disappointed" by Mr Swinney's stance, and insisted "only collective discussions at a national level can resolve the issue".
The row between the Scottish Government and local government was sparked after the Budget for 2015-16 - which was passed by Holyrood on Wednesday - included a further £10 million for councils to help them maintain the number of teaching staff in Scotland's schools, up from the £41 million originally planned.
But Mr Swinney said despite having had talks with Cosla he had been unable to reach a deal with leaders, and as a result he said the Scottish Government had "no alternative to make that funding available on a council by council basis, if and only if councils are prepared to sign up to a clear commitment to protect teacher numbers".
The Deputy First Minister warned that a "failure to deliver" would "result in a claw back of funding".
Teacher numbers fell last year, with the ratio of pupils to teachers rising at the same time, despite a commitment by the Scottish Government to maintain staff levels in the classroom.
Council leaders discussed the situation at a special meeting in Edinburgh today.
Afterwards, Mr O'Neill said: "I was given a very clear and constructive message from council leaders today. They instructed me to write to Mr Swinney in no uncertain terms calling upon the Scottish Government to re-engage in national discussions on the issue of teacher numbers."
He added: "Scotland's council leaders were clear that the Government's proposal to bypass national discussions with Cosla was totally unacceptable and not something that they would tolerate.
"They were very disappointed in the tone and stance of Mr Swinney on this issue and particularly the assertion that Cosla had walked away from a negotiation. Another source of anger is the threat of financial sanctions.
"Council leaders were adamant that only collective discussions at a national level can resolve the issue and that no council be sanctioned as a result of workforce planning issues.
"To put it simply, Scottish Government tried to bypass Cosla on a difficult and complex issue by taking national negotiation off the table. Council leaders put it right back on today."
David Ross, the Labour leader of Fife Council, claimed Mr Swinney "seems intent on demonising local authorities for something that is largely beyond their control".
He added: "In Fife we face the prospect of having to recruit an additional 25 teachers next year because of increasing pupil numbers. These teachers just aren't there. And now it seems we are being threatened with a loss of grant for something that is not our fault."
Labour finance spokeswoman Jackie Baillie also urged Mr Swinney to get round the table with Cosla leaders.
She said: "Scottish Labour is committed to maintaining teacher numbers, unlike the SNP who have presided over the removal of 4,275 teachers from our classrooms.
"The SNP Government has come in for some well-deserved criticism from council leaders today. The Deputy First Minister John Swinney's crude attempt at divide and rule has failed abysmally.
"The SNP are responsible for Scotland's teacher shortages as has been highlighted by Fife Council earlier. He can now expect letters from all Labour's council leaders asking the same question, which is how can they meet his teacher targets when there aren't enough teachers to fill vacancies?
"Instead of issuing threats Mr Swinney now needs to get round the table and negotiate a proper financial settlement for Scotland's councils. He might want to take Education Secretary Angela Constance with him to explain how she's going to fill the gap in so many of Scotland's classrooms."
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