The lower house of parliament's justice commission approved the draft law last week and it is due to be approved by MPs by the summer.
It would allow euthanasia, already legal for adults, to be available to terminally ill children in unbearable pain if they request it and have permission from their parents.
The four bishops said in a statement they wanted to “wake up people’s consciences”.
They invited those of other faiths as well as atheists to support their campaign because their objections, they said, were philosophical rather than strictly religious.
All deaneries, basilicas and shrines in the archdiocese were also encouraged to organise their own fasts and prayers.