Unite said drivers on strike had been told during talks their pay was likely to go down from £40,000 to £32,000.
It stated: "For WRCO workers who do not wish to make a sideways move, the council are saying in public that they will get a one-off payment of £16,000 which would cover two years' loss of £8,000 in pay cuts.
"Again, if this is true this needs to be put in writing."
General secretary Sharon Graham said she stood ready to meet Cotton and deputy prime minister Angela Rayner over Easter.
But she said the council leader should "rethink his position" if his comments "prove again to be untrue".
"We appear to be in a parallel universe. Yet again John Cotton is saying one thing in public, while his local officers are saying another in the negotiating room and in writing," she said.
On Tuesday, the council said the amount of uncollected waste had peaked at 22,000 tonnes, and it was on track to clear the backlog by the weekend.