The city's transport convener, Councillor Gordon Mackenzie, said the "ludicrous" position adopted by German firm Bilfinger Berger meant the city now had no option but to terminate the contract.
It is likely the move will lead to a legal dispute which will see all work in the city grind to a halt and set back the opening of the tram line by two to three years.
The move comes as the Evening News walked the 11-mile route and found little evidence of any progress being made on a project which had originally been due to open next year.
Cllr Mackenzie said it was now necessary to "call time" on the dispute with Bilfinger, which first began in February last year.
He said he would give the construction giant just three weeks to change its position before tram firm TIE began breach of contract procedures.
He said: "This city will not be held to ransom by this contractor any longer.There would be worse things for the project than for this contract to be terminated.
"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."
He praised the German firm's consortium partners Siemens and CAF, but criticised Bilfinger for its "can't do" attitude.
He added: "The utility diversions are 97 per cent complete. However, the contractor has insisted that they require an additional 30 months to complete works. This is ludicrous.
"Every major infrastructure contract includes normal design development and while we accept that there have been substantial changes to the design, the claims that this contractor are putting in are from the realms of fantasy.
"If we don't get an acceptable cost and programme soon I won't be able to recommend the council continues its support for the project with this contractor. We are at the brink. Personally, I'm prepared for that and I also hope the consortium recognise it as well, but I'm not optimistic."
At the weekend Cllr Mackenzie also cast doubt on the project keeping to its 2012 deadline, saying that the dispute with Bilfinger meant that "as every month goes by, you can add a month to the completion date".
The council, which must finalise next year's budget proposals by September, is now in a position where it may have to pay Bilfinger to quit the job, or embark on a lengthy and costly legal battle.
A similar dispute in Vancouver saw Bilfinger kicked off a project to build two underground tunnels. The row set back that project and is said to have added $200m on to the bill.
The best hope for Edinburgh now is that at least some part of the line can be opened with the money available, with some senior sources now admitting they are ready to talk about "phasing" the route.
David Mackay, chairman of Edinburgh Trams, said: "I totally understand Cllr 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."
Council officials were today reported to be involved in a row with tram firm TIE over whether to disclose the final costs of the project.
Finance director Donald McGougan was said to be keen to spell out the extent of the cost overrun, paving the way for the council to borrow the extra funds.
But TIE is understood to be arguing such disclosure would damage its position in contractual negotiations.
Tramline's in a whole lot of rubble
Trams dispute: 'It's a brave move – and a huge gamble'