A full meeting of the city council next week is due to vote on whether to go ahead with a £144 million proposal to complete the route to Newhaven.
But the SNP group has voiced doubts about the financial case for the extension at a time when the council is under massive budget pressures.
A senior SNP source said the group supported taking the trams down Leith Walk in principle, but did not believe the current case was robust enough, despite a recommendation by officials in favour of the move.
The source said although they had not taken a final decision, the Nationalists were likely to vote against any extension at this stage.
Their stance puts their Labour coalition partners in a difficult position. They must decide whether to try to press ahead with the project regardless of the SNP views, effectively splitting the coalition with 18 months to go until the next council elections, or accept that the much talked-about tram extension should be put on hold for now.
The SNP source said: “We would like to see the tram extended, but this case is not robust enough to justify it.
“At the moment, it only stacks up in the best scenario, but it needs to add up for all scenarios. We need to be clear the tram could be funded without putting additional pressure on the council’s already squeezed budget.”
The source predicted Labour would recognise the need for as wide a consensus as possible in favour of an extension before it tried to go ahead with the scheme.
Conservative, Green and Liberal Democrat groups have yet to decide their positions on the proposed extension, which is due to be debated on Thursday next week.
Three options are on the table – completing the route to Newhaven, the end point of the original route, at a cost of £144.7m; taking the line to Ocean Terminal for £126.6m; and stopping it at the foot of Leith Walk, priced at £78.7m.
Officials have recommended going all the way to Newhaven.
Councillors have been given restricted access to confidential data on the project to help them come to a view ahead of the council meeting. The tram project – dogged by delays and cost increases – had a final price tag of £776m when services began running 18 months ago.
A total of 4,920,000 passengers were carried in the first year of operation – eight per cent above target – and customer satisfaction has been recorded at 95 per cent.
Transport leader and Labour councillor Lesley Hinds declined to discuss what view her group might take.
She said: “Both Labour and SNP groups are part of the Capital Coalition. Any decision will look at the finances involved and the investment in the future of the city. I’m sure both groups will take that into account.”