The row between a consortium including the German firm Bilfinger Berger and tram project managers Transport Initiatives Edinburgh (Tie) has resulted in work to lay tram tracks being put on hold.
The closure of the main shopping thoroughfare went ahead on Saturday as plans to re-route traffic to accommodate the closure of Princes Street were too far advanced to be halted.
Contractor consortium BSC claim they can't start work until an agreement has been reached on additional costs.
Tie said: "The project was given 48 hours notice before the commencement of works that the consortium, BSC, comprising Bilfinger Berger, Siemens and CAF, wished to impose unacceptable conditions as a requirement."
According to Tie these conditions could add £50m to £80m to the cost of the project.
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Bilfinger Berger said utility works, including the routing of electricity, gas and water supplies, were fallen a year behind schedule, leading to an impact on costs.
Edinburgh Conservative MSP David McLetchie told BBC Radio's Good Morning Scotland programme: "This needs to be resolved this week, it needs to be resolved quickly.
"That should be possible in terms of the contract but I think that the council has to stand very firm on this and some of the figures that are being bandied about like wanting an extra £80m are frankly ridiculous.
"There can be no question of the council or the Scottish taxpayer, for that matter, being held to ransom on this."
Steve Cardownie, deputy leader of Edinburgh City Council, said: "The best scenario for everyone concerned would be for Bilfinger to drop their demands and commence with their work. There is an issue regarding our future working relationship with them because of what has happened.
"Unless it can be justified, no further money should be given to the company. We cannot stand by and let people plunder the public purse."
The closure of Princes Street was planned until the end of November although if no agreement can be reached it will reopen next week.
As the dispute continues of a tram the city of Edinburgh council have placed a mock-up of a tram for members of the public to view.
The mock-up was produced as part of the contract by the Spanish manufacturer, CAF, for necessary testing and approval by authorities.
Once up and running, there will be 27 trams in total. Each tram is 42.8m long and 2.65m wide with 7 articulated sections. There is a capacity for approximately 250 passengers on each, which is the equivalent to 2.5 double deck buses.
(GK/JM)
