Council chiefs are considering a range of designs, which would see two trams given a radically different paint job from those in the normal company livery.
The proposals are at a very early stage, but other suggestions have included a gold tram and turning one into a "Santa Express" at Christmas.
Dave Anderson, director of city development, said he hoped that Edinburgh's trams would come to be readily associated with the city in the way red Routemaster buses are with London and yellow taxis with New York, making it important to project the right image.
He said: "Edinburgh is a very elegant city, and we want the trams to be sleek and elegant. They need to have the capacity to surprise and delight people, and we need to think about what we want people's first impression of the city to be. We're thinking about what kind of experience customers can have on arrival at the airport.
"There's an opportunity to do something with the real design talent we have in Edinburgh. We want to see people think so highly of the tram system that they get photographed beside it – that it becomes an icon."
The council has asked a city design consultancy to come up with ideas for what the trams could look like.
Among the designs shown to councillors at a tram committee meeting was one based on a tartan pattern.
But Graham Scott, creative director of Edinburgh-based Nevis Design Consultants, said using tartan would perhaps project the wrong image of the city.
He said: "It depends how it's implemented. If it's treated in a contemporary way it could work, but not if it looks like the sort of tat you get on the Royal Mile.
"They have a real opportunity here to make a statement, and you just wonder if they will. I must say that the designs I have seen so far are a bit disappointing. They have just taken the bus livery and stuck it on the trams"
Earlier this week it was revealed that bosses hope to display a full-scale model of an Edinburgh tram early in the new year, as they bid to win over opponents of the 512 million project.
It is believed that one idea currently under consideration is to create a mock-up of a tram stop, which could go on show in Princes Street Gardens. More than 24,000 people visited a replica of a tram in the Gardens in 2005.