|Your best bet is to cut a chunk of the vine out and pry it off the tree. It will come - trust me I did this for a number of vines climbing pine and cedar trees in my yard.
If you use anything that is systemic, you risk damaging the trees. Herbicides work best when you can spray on the leaves which are likely 20 feet up. The instructions for RoundUp Poison Ivy mentions that if vines are taller than 5 ft. that you need to cut down and then spray.
What you see as deeply embedded is really more likely the poison ivy rootlets grasping tightly to the host tree. It can resemble a thick hairy mess. If you cut the vine, do not use power tools as this can spray the urushiol oil. I found using a bow saw and cutting carefully worked best.