Assembly, Gilded Balloon, Pleasance and Underbelly, who produce their own joint guide every year, have insisted they have merely “suspended” their programmes for this year.
It is thought they would stage much smaller-scale line-ups than planned if social distancing restrictions are lifted in Edinburgh in June or July.
They issued a joint statement to companies and performers ahead of an official statement declaring that the five main August festivals were off.
The four operators pledged to “give it our best shot” to stage their own programmes in August if they had the approval from the various authorities, but also admitted they would also need the “wholehearted” backing of the public.
The Fringe Society, which insists that the festival will “not go ahead,” announced its decision jointly with the International Festival, the Tattoo, the art festival and the book festival.
Chief executive Shona McCarthy said the decision not to go ahead with this year’s event was “unavoidable” in light of the current circumstances, adding: “Public health must and always will come first.”
But in a Q&A posted on its website, the Fringe Society admits it “does not have the power to cancel the festival as a whole.”
It adds: “The Fringe remains an open access festival, which means the Fringe Society does not decide who can and cannot put on shows.
“We are advising all venues and companies to follow the latest government and public health advice, and will continue to provide support and guidance for all participants as the situation progresses.”
The statement from the four companies said: “Whilst we remain hopeful that we will get through the worst of this crisis by August, over recent days it has become increasingly clear that there will be a great many uncertainties.
“A definitive national timeline for recovery and long-term policies on public gatherings are understandably some time off. Without these, it is with great sadness that we have taken the difficult decision to suspend planning for our 2020 Fringe operation. At this point, each of our venues will refund all tickets to the general public and take shows off sale both at our own box offices and with the Fringe Society.
“Whilst we are suspending our activity for the foreseeable future, if there is any chance that we might rekindle the spark of a Festival Fringe at our venues in August 2020 and rebuild an event for this summer, we will certainly try - for all those artists who want to show their work and for the whole festival community who depend on the Fringe, as well as for Edinburgh as it seeks to rebuild after the lockdown.
“Obviously, any potential to rebuild our festival would have to have the sanction of national and local authorities, the health service, the emergency services and our landlords; and, most importantly, it would need the wholehearted support of the public and our artists.
“The undefinable and indefatigable Fringe spirit has always been one of this festival’s most exhilarating characteristics and if there is any way that we can capture just a tiny ounce of that spirit this summer, then we hope you know we will give it our best shot.”
A statement posted on the Pleasance website said there was “no current alternative” to suspending its planning for this year.
However it added: “In the future, when restrictions are lifted and life returns to some kind of normality, we will hopefully reopen, rebuild and once again capture that undefinable Fringe spirit that embodies all those that perform with us, work with us and visit us.
“Our London theatres doors will open as soon as we’re able and we’ll return to the Fringe galvanised within a renewed purpose, be that in 2020 or 2021.”
Assembly told its Twitter followers that it was “suspending our 2020 plans for now.”
It added: “Assembly has provided a stage for new and established artists at the world’s greatest arts festival for the last 40 years and plans to do whatever it can to continue to do this in the future.
A further statement on its website added: “Whilst the immediate future is uncertain, we are optimistic that the pioneering spirit of the Fringe will continue.”
Underbelly directors Charlie Wood and Ed Bartlam said: “We sympathise with the artists who were working so hard towards the Fringe and who are facing such immediate threats to their livelihood, and will do what we can to support them.
“We realise that the festival is important to so many people, but the safety of the participants, audiences, people of Edinburgh and the staff both year-round and temporary must come first.
A statement from Gilded Balloon said: “For the past few weeks, in conjunction with the Fringe Society and all our festival partners, we’ve been working hard to explore all avenues to put on this summer’s festival.
“However, due to the continuing spread of the coronavirus, the increasing threat to people’s health, and the potential need for an extended period of social distancing, we have taken the collective decision to suspend our 2020 programme as it stands.
“This means that as of today we will be taking shows off sale and automatically refunding any ticket purchases made for the 2020 Gilded Balloon programme for this Fringe, less booking fees.
“We are heartbroken to have had to make this decision – particularly for the artists who have made such great work and for all those people for whom the festival is so important.
“We’ll be in touch once we have more news but for now, stay safe and look out for each other.”