The decision to build the line as far as St Andrew Square will be confirmed at the next meeting of council on 25 August, pending further information on costs, timescales and associated risks - and ahead of the 31 August deadline agreed as part of the mediation process with contractor Bilfinger Berger.
The agreed option takes the project cost to £773m, more than £200m above initial budget.
Other options set out in a report to the council were to scrap the whole project, at a cost of up to £750m, or stop short of the city centre at Haymarket, but this would have saved only £73m and would have required an annual subsidy of £4m a year to cover losses.
The chosen option of continuing to the city centre is expected to generate £2m a year in revenue.
The full scheme, as originally envisaged, took the line from the airport to Newhaven – a further 5km – with initial budget set at £545m.
Council leader Jenny Dawe said: "I believe that a first phase to St Andrew Square is absolutely the right decision for moving forward with the project. I look forward to the further report from the chief executive later in the summer, which I believe will strengthen the case for building to St Andrew Square and beyond - whilst giving more clarity on funding.”
Chief executive Sue Bruce said: "As part of the plans for moving forward, it was agreed that the programme of works to remediate and complete the Princes Street section of the route could go ahead at the beginning of September. Full plans for this programme will be released shortly."
So far, £461m has already been spent, including £47m on trams. Construction disputes are the subject of mediation. Last month it was agreed that Bilfinger Berger and Siemens (BBS) would carry out a programme of remedial works to repair the road surface. The works will be carried out in two stages: from 3 July - 26 November and then from January - May next year. The contractor is bearing the cost of this.