A loan deal of about £80 million could be enough to bankroll the line from the city centre through Leith and as far as Newhaven.
City bosses approached Henderson Global Investors (HGI), who boast £70 billion worth of assets, in the hope of securing a deal cheaper than a typical repayment plan.
Extending the tracks is seen as a way of making a profit. Repaying loans on a successful line is seen preferable to subsidising a loss-making system.
The extension could benefit HGI by attracting commuters and customers to the city's St James Quarter, which the firm is developing for £850m.
A source close to the talks said yesterday there were "benefits for both sides in such a deal". The source said: "It makes no sense for the council to subsidise a tram route when the money could be better spent securing and paying for a more profitable route.
"For the developer having a tram line down Leith Walk has always been a key feature to the success of the development.
"At present the city has a largely commercial route to the airport. This would create a more profitable residential route."
Council bosses will set £3.2m aside each year to cover the start-up costs of the current tramline and predicted losses.
The council is repaying £15.3m a year as part of a 30-year payment plan borrowed at a 5.1 per cent interest rate. A £231m rescue loan was signed in 2011 to help bankroll the curtailed 8.7-mile line from the airport to York Place.
It means the city is paying out £228m in interest under the terms of the loan, tipping the tram budget over the £1bn mark.
It has been estimated taking the trams the extra 2.85 miles to Newhaven would cost £80m. A lot of the materials - such as tracks and tram cars - have been already bought and are in storage.
But Transport Minister Keith Brown has warned there would be no more Scottish Government funds for the network.
City transport convener Councillor Lesley Hinds confirmed a committee report on the proposal could be ready by the end of the year. She said: "We have always said the next stage would be to finish off the line.
"We have got the rails and the trams to go down to Newhaven, but for this to go ahead, we need public support and we also need resources."