The move is being considered to accommodate demand from performers and ease pressure on the Royal Mile and The Mound precinct.
It would be the biggest expansion in its official street areas for around 20 years if it went ahead.
It is being planned in the wake of an overhaul of the long-running street theatre arena on the High Street section of the Royal Mile to create better stages, ease crowd congestion and accommodate large-scale performances for the first time.
The Fringe Society is exploring options for expansion at the same time as council chiefs are considering various “traffic free” experiments in parts of the city centre, including the New Town.
It may also face restrictions over the use of The Mound precinct in the next few years due to planned building work on an expansion of the Scottish National Gallery complex.
The Fringe’s plans have emerged a year after the organisers of the Edinburgh International Book Festival called for the whole of George Street to be made traffic free in August. The event spilled out from Charlotte Square Garden onto part of the historic thoroughfare last year and this month has seen one of its biggest venues moved into George Street, alongside a cafe-bar and bookshop.
St Andrew Square Garden was previously home to Fringe shows, but these were controversially ousted last year amid concern over the impact of infrastructure on the grass.
Other events have been allowed to go ahead by the square’s owners since then in the space, including an ice rink during the city’s winter festivals and free screenings in the run-up to the Edinburgh International Film Festival.
Fringe chief executive Shona McCarthy said: “We have 250 street performances a day at the moment, but it is clearly an area of the Fringe that could expand.
“We’re really up for open conversations about any aspects of the public realm that we could be involved in animating or enhancing in any way. We’re really interested in the New Town area, particularly George Street and St Andrew Square.
“We’re keen to work with anyone to look at what kinds of performances would work in specific spaces, what would be in keeping with particular areas and what would work for local businesses around those spaces. One of the things that is under-represented is some of the amazing circus acts and aerial performances that are at the Fringe.”
Ms McCarthy said the Fringe was hoping to take a “fresh view” of its use of The Mound precinct in the near future.
She added: “It feels to me like The Mound is a bit squeezed at the moment for performers and audiences. There are bottlenecks where there could be a much better flow of people.”
Roddy Smith, chief executive of Essential Edinburgh, the business group that acts as a go-between over the staging of events in the New Town, said: “We certainly have ambitions to see George Street and Castle Street used as much as possible for festival activities, but it will be really important to have a real mix of things, including street theatre.”