A report to councillors next week is also expected to confirm tram bosses are discussing whether to end their contract with Bilfinger Berger because the project is so behind schedule and over budget.
The report is understood to warn that the £545m project still faces major problems, including the dispute with contractor Bilfinger Berger.
The city tram company, Tie, has not asked for extra funds yet. But it is understood the council is exploring contingency plans to borrow up to £55m to pay for the cost over-run.
Earlier this month the City Council gave the German giant a three-week deadline to find a solution to a raft of problems on the job.
The project was originally due to finish in August 2011. It is now seven months behind schedule and and running well over budget.
Gordon Mackenzie, Edinburgh City Council’s transport convener, said: “This city will not be held to ransom by this contractor any longer.
“There needs to be a move to settle the project’s contractual dispute within the month.
“There would be worse things for the project than for this contract to be terminated.
“The management team at Edinburgh Trams have done what they can to find a reasonable way through this dispute, but now it’s time for action.
“Week after week I’m told of excessive claims, month after month progress falls behind schedule, time after time they’ve been told a cost plus deal is not on. The people of Edinburgh deserve better.”
David Mackay, chairman of Edinburgh Trams, said: “I totally understand Councillor Mackenzie’s position.
“We have been hugely frustrated by the lack of progress in settling this dispute and the subsequent lack of progress on the ground.”