The proposal for trees along the Sabbath Walk in the Dale Coppice at Coalbrookdale comes from the Severn Gorge Countryside Trust (SGCT).
The plans have been significantly scaled back from a previous proposal that attracted vocal opposition and significant criticism from a group of nearby residents, who argued it would ruin "a beautiful path".
Under the latest plan 47 trees will be subject to the work, with 13 of those set to be felled completely.
Other work will include removing some dead wood and some tree limbs, while other trees will be 'monolithed', where all their branches are removed and the trunk remains.
The trust said that the majority, if not all, of the timber felled will be left in the woodland to break down as part of the natural carbon cycle.
The previous plan could have seen nearly 140 trees chopped down, and residents had been particularly frustrated after red marks appeared on trees which were due to be felled.
The trust said it had listened to the public's concerns and decided to scale back the work, which was now focussed solely on public safety.
The organisation said the work is needed because damage to the trees caused by squirrels is putting residents at risk of being hit by falling branches.
SGCT chief executive John Brayford said: "Following on from the public meeting and site visits in October last year the Severn Gorge Countryside Trust has listened to and considered the comments from local residents and has reduced and adjusted the approach to the proposed tree works in Dale Coppice.
"The recent planning application submitted to Telford & Wrekin Council is to work on 40 trees in Dale Coppice and is based solely on public safety following advice from our arboriculture consultants.
"The SGCT appreciates the love of, and concerns for, woodlands and open spaces demonstrated by the local community, but these works have to be undertaken as part of our commitment to health and safety.
"Further consultation with the local community on the future management of trees and woodland within the World Heritage Site is being planned for autumn 2023."
A decision on the proposal will be taken by Telford & Wrekin Council at a later date.