Supporters have until January to come up with a plan
Amsterdam city council ruled in March that the rotting 150-year-old tree must be felled as a danger to the public.
Following protests the council has given those who want to save the tree until January to come up with a plan.
The tree was a ray of hope for the famous diary writer as she hid in the attic of the canal-side warehouse.
The Jewish teenager remained indoors with her family for 25 months until they were arrested in August 1944.
She died of typhus in the Bergen-Belsen camp in March 1945.
The attic window from which Anne Frank could see the tree was the only one that had not been blacked out.
In an entry dated February 23, 1944, she wrote: "From my favourite spot on the floor I look up at the blue sky and the bare chestnut tree, on whose branches little raindrops shine, appearing like silver, and at the seagulls and other birds as they glide on the wind...
"As long as this exists, I thought, and I may live to see it, this sunshine, the cloudless skies, while this lasts I cannot be unhappy."
Ton Boon of the Amsterdam Centrum borough told Agence France-Presse news agency there was "only one Anne Frank tree" and it had been agreed to allow time for a possible rescue plan.
Experts say the 27-tonne tree is too diseased from fungi to be saved and the owner wants it cut down as he would be liable for any damage caused should it fall.
The tree is adjacent to the building that now houses the Anne Frank Museum.
A Utrecht-based firm, Trees Institute, has suggested a salvage plan involving treatment and support for the trunk and limbs.Spokesman Edwin Koot told Associated Press: "The tree represented freedom... to Anne Frank. We must go the extra mile to try to save it."