Partners have been holding talks about the arts festival taking place in August amid safety fears around the Covid-19 pandemic.
Many large-scale events - such as Euro 2020 - have already been delayed due to the coronavirus outbreak.
Fringe Society chief executive Shona McCarthy said: "I'm grateful to the hundreds of artists, producers, venues, residents and audience members who have been in touch with us this week to share their thoughts on the future of this year's Fringe.
"Whilst there have been many different opinions expressed, we know that definitive answers are needed and we hope to have them for you next week.
"Progress is being made in this area and we will continue to push for answers over the coming days."
She added: "This week we have been in regular dialogue with Scottish Government, City of Edinburgh Council, venues, partners and other stakeholders to work through all the options and find solutions.
"This is a complicated process but one that is moving forward daily.
"I appreciate that the delay is frustrating but I promise we will do everything we can to update you as soon as possible over the coming days.
"In the meantime, please keep talking to us."