More than 700 people have signed a document urging the local authority to freeze the Flood Prevention Scheme, which is being implemented after floods in April 2000 caused tens of millions of pounds in damages.
Many of the trees have large roots, which are said to be obstructing the construction of flood walls.
However, many residents say the environment and natural beauty of the area is being ruined and have urged the council to call a halt.
The council and its contractors held a heated public meeting at Stockbridge Library on Wednesday night, at which residents said they had not been consulted on the project.
Petition author Ani Rinchen Khandro, who lives on the banks of the river, said: “We first heard about this around three years ago and we wrote a letter to the council to raise our concerns.
“Although we didn’t get a reply, that was the last we heard of this scheme until now.
“Then last week I got a note saying they were going to start cutting down the trees. It’s really quite surprising as there hasn’t been consultation of any kind.
“We do appreciate some flood prevention has to be implemented, I live there and have an interest in not being flooded, but where we disagree is the method. It’s a very heavy-handed approach. A sledgehammer to crack a nut.
“It’s a local beauty spot, thousands of people go past every day, including tourists on their way to the Botanic Gardens, so it’s really quite tragic.
“These trees are nature’s flood prevention, as their huge roots soak up water and maintain the riverbank, so we believe it’s counterproductive to cut them down like this.”
Jim Henderson, 63, a retired lawyer, said he regarded the scheme as “vandalism”.
He said: “I went along to the meeting and the council’s engineer basically told us that the meeting was just to let us know what is happening and that the work will go ahead either way.
“He also explained to us that the floods of April 2000 were a one-in-a-200-year event, which left us wondering why such drastic action must be taken.
“It’s just a dreadful act of vandalism and it’s actually quite depressing to walk down there and see the trees being felled.”
Work began on site for the Water of Leith Flood Prevention Scheme in March, although much of the tree felling has begun in recent days.
Councillor Gordon Mackenzie, transport convener at the city council, said: “It’s always regrettable when trees need to be cut down, but we simply cannot construct substantial enough foundations for the flood defences while these trees remain.”