Councillor Gordon Matheson, Labour leader of Glasgow City Council, has written a vitriolic letter to Swinney claiming his local authority's budget will be cut by 13.5 million more than other councils.
Matheson was reacting to the Scottish Government's 11.5 billion local government grant settlement for the next financial year, which will be allocated on the condition that councils implement key SNP policies such as a council tax freeze, maintaining police numbers, free personal care and cutting teacher unemployment.
In his letter, Matheson argued Glasgow was being cut more ferociously than other local authorities but he has to accept the SNP's demands or risk his budget being cut by a further 51m.
"I believe your behaviour towards local government to be well below the standard I expect of a government minister," Matheson said. "You are now bullying us. Your government has fallen a long way from the high ground of 2007."
Matheson described the conditions set out by the SNP administration as a "threat", adding: "I regard your tactic as short-termist political blackmail."
Matheson argued that the money allocated for Glasgow was a "smaller share of a smaller pot".
He said: "The average cut to councils' budgets is 2.6 per cent. Glasgow's cut is 3.6 per cent, or in cash terms a further 13.5m... To be clear, I regard this as the SNP's extra cut for Glasgow."
Matheson said the SNP's plan to freeze council tax was "fool's gold" that would eventually have to go.
Swinney's spokesman said: "The agreement negotiated between the Scottish Government and the Convention of Scottish Local Authorities represents the best possible deal for local councils and people in the face of massive Westminster cuts.
"It provides all councils with a greater degree of protection than other parts of the public sector and is far superior to the disastrous plight facing local authorities in England."