Fergus Linehan said the event is facing the nightmare prospect of being unable to get artists, companies and even sets for shows to the city in time if travel delays and problems in securing visas and permits start piling up.
Mr Linehan said the worst possible scenario was that the festival faced “overnight” changes imposed by government departments.
However the Irishman said he had programmed his most international line-up of artists during his five-year tenure with the event, which will return to Leith Theatre and expand into Tynecastle Park football stadium and Jupiter Artland sculpture park for the first time.
Around 2600 artists from 40 nations are represented in this year's line-up, including Australia, Nigeria, Canada, Belgium, China, Mali, Holland, South Africa, New Zealand, France, Germany and India,
This year's programme includes a strand exploring what “local, national and international citizenship might mean in the 21st century.”
Mr Linehan said: “We have to act as if everything is going to be okay, or we wouldn’t be able to put the festival on.
“Edinburgh is the busiest tourist destination in the UK after London. Everything has to run like clockwork for us.
“We’re now getting down to the technical questions like ‘will we be able to get all of the equipment here for a show?’ or ‘will there any any disruption to flights?’
“A catastrophic no-deal Brexit which caused problems in relation to travel is definitely not to be desired. Even before Brexit, it was very hard to transfer through Heathrow Airport last summer.
“The worst-case scenario for us is any delays to transport links. That applies to all the other festivals.
“If government departments have additional layers of pressure on them and their systems slow down it would be really testing. Any changes in arrangements that create extra layers of administration that would be extraordinarily difficult. Everyone is in the same boat on this.
“You can only plan on the basis of trying to be as careful as possible and hope everything will be fine.
“You would hope that regardless of what happens people will realise that we will need time and we don’t go from the status quo to new arrangements overnight.
“We are an international festival, we have got to produce an international programme and, if anything, this is the most important year to be international.”