Tram firm TIE, which is already building the Newhaven to airport 1a line, has launched a project team to investigate ways of paying for the spur from Roseburn to Granton, and say they are "optimistic" it will go ahead.
The 1b link was originally put on hold when the tram scheme was scaled back in 2006.
However, TIE has secured a fixed price of 87 million to build the line – if it can commit to the project before next spring. TIE now faces a race against time to get funding in place.
The 543m already in place to build 1a includes 96m in "contingency" funding – but it is thought only around half of this will be left to put towards the spur.
Among the options being considered for any funding gap are selling off council assets, pressing for more developer contributions and borrowing from either public or private sources.
Opposition politicians today said they would be wary of selling off council assets to fund line 1b, but council and tram chiefs said they were committed to finding a way of making the project happen.
Willie Gallagher, executive chairman of TIE, said: "We plan to be back in front of the councillors around September with our view on the likely funding that is left from the 543m available.
"If it is, say, around 37m that needs to be raised, then you've got to remember that is a small percentage in the context of a 543m project.
"We have put a project team together to look at the funding options for 1b, such as realising developer contributions.
"Obviously the focus is on line 1a just now, but we are optimistic that we will be able to get a package in place for 1b, which does offer fantastic gains not just for the waterfront, but also Granton."
It is hoped phase 1b would be able to open by the end of 2012 – a year-and-a-half after the rest of the first line is completed – if a decision to proceed with the work is taken by city leaders this year.
Councillor Mark McInnes, the city's Tory transport spokesman, said: "We would be wary of any attempts to sell off the family silver to try to pay for 1b.
"It is vital that we get value for money, whatever they are considering. Selling council assets or going for any sort of leasing arrangement may not present the best option.
"Increased developer contributions are the obvious attractive way of filling any funding gap, particularly given the benefits that this line will bring them."
The Roseburn to Granton spur has been a controversial part of the proposed tram network, not least because it would run along the Roseburn Urban Wildlife Corridor.
Councillor Phil Wheeler, the city's transport leader, said: "We are still absolutely committed to finding a funding solution to ensure line 1b of the tram route is built as quickly and as cost-effectively as possible."