A deal between striking bin workers and Birmingham City Council could have been reached were it not for the interventions of the Unite union's national leadership team, the BBC has been told.
A message seen by the BBC also appears to suggest some local union members are dissatisfied with how the dispute has been handled.
Unite said claims national union figures had scuppered a potential deal were "entirely without merit".
Speaking on a visit to Birmingham on Thursday, Deputy Prime Minister Angela Rayner urged the union to accept what she said was a "significantly improved" offer from the council.
Thousands of tonnes of rubbish have built up on the city's streets over more than four weeks and Ms Rayner said the situation was "causing misery and disruption to residents".
A number of senior Labour figures outside the council, including some MPs, said they understood the purported deal with local reps would have been reached during the rolling strike in January and February.
However, individuals with knowledge of the situation said while the alleged deal was backed by refuse workers at two of Birmingham City Council's three bin lorry depots, National Unite figures were involved at the third site where it was rejected.
Sources, including some with extensive union experience, say the handling of the dispute is being influenced by power struggles within Unite.
A longstanding Unite member from the West Midlands told the BBC that while commissioners at the council bore some of the responsibility for the dispute, the regional office of Unite was unhappy it had been bypassed.
They said: "Unions traditionally have been about the art of the deal on behalf of their members. Instead, Unite and some others seem to be in the grip of people for whom disruption, disputes and revolution are their priority."
The industrial action, which began in January and escalated to an all-out walkout in March, has seen hundreds of Birmingham's refuse workers on strike.
It centres on the loss of a job role that bin workers describe as "safety-critical", which Unite said would affect about 150 workers and mean a cut of up to £8,000 a year for some and the loss of pay progression for hundreds of others.
Birmingham council says the post does not exist at other authorities and disputes both the number of staff affected and the sum of money involved.
Councillors have previously said that no worker "need lose a penny".
Ms Rayner, who is also Communities Secretary, said she wanted to see "a fair resolution to this" and added: "I would urge Unite to suspend the action and accept the improved deal."
She said the council had "moved significantly to meet the demands of the workers so we can see an end to this dispute".
She also said the backlog "must be dealt with quickly to address public health risks".
