Talks between the Labour-run council and Unite have been taking place with conciliation service Acas since the start of May, after previous negotiations to resolve the strike ended without a solution.
The union's latest statement accuses the council of failing to meet a promised deadline to lay out an offer – and questions the very existence of such an offer.
"From the council side, the negotiations in this dispute have been a shambles, with the government right at the heart of it," said Unite's General Secretary, Sharon Graham.
She called on council leader John Cotton to "stop playing games, get in the room and solve this dispute".
"The bottom line is that our members can't afford to have savage pay cuts of up to £8,000 with no mitigation," she added.
Ms Graham told BBC Radio 4's Today programme that Unite had expected a revised deal to be presented to them on 8 May but then learned it needed to go to the government commissioners to be signed off.
She added: "That means the council itself is not the final decision maker in this process – it's actually the government and we're calling on the government to get that offer on the table so we can see it and see if it's good enough for our members."
Ms Graham criticised Mr Cotton for not attending negotiations and said Unite wanted the "fair and reasonable offer" referred to by Mr Cotton, Sir Keir Starmer and Angela Rayner, though she claimed that offer "simply does not exist".
"It's totally and utterly unacceptable – nobody can take that sort of a pay cut. For some of these people, it's a quarter of their pay," she added.
Ms Graham later told BBC Radio WM that the process was like "blancmange".
She said: "I feel appalled for the people of Birmingham because of how they've been let down."
A spokesperson for the council said the authority was committed to seeking a negotiated settlement to end the dispute.
However, in a message sent to Unite via the conciliation service Acas last week that was seen by the BBC, the authority said "it would not be in a position to make further offers to settle this dispute until we have cleared all necessary commissioner and constitutional approvals and we anticipate this will take up to seven days".
'Monumental governance disasters'
Council sources told the BBC's Rob Mayor that did not amount to any blocking of a deal as claimed by Unite, but simply that there needed to be time for proper due diligence into any proposals.
"They stress that the authority is an organisation which has seen it's reputation shredded by a series of monumental governance and financial disasters, leading to it's effective bankruptcy in 2023," the sources said.
"They say it would be foolhardy not to follow measures, like the oversight of government commissioners, which were put in place to ensure those mistakes are not made again."
They said: "Our focus has been to find a solution to this that does not put the council in a position that compromises us financially or legally. This is why we are committed to making a revised offer."
It also defended the commissioners and said they were also "fully supportive" of finding a solution to the strike.
A Ministry of Housing, Communities and Local Government spokesperson called on Unite to suspend strike action and urged both parties to "reach agreement on a fair and reasonable offer".
