Dr Salisbury said the stretch of coast was "immensely important" in national and global terms.
"I don't know for sure yet whether they have commenced drilling in that area, but their maps indicate that they intend to," he said.
Broome-based dinosaur footprint tracker Louise Middleton, who with late partner Paul Foulkes brought the James Price Point tracks to scientific attention in the mid-80s, said she was "horrified" by the prospect.
"This site is recognised as having one of the most significant concentrations of preserved dinosaur tracks in the world," Ms Middleton said.
"What makes it special is not only the number of track, but that there are so many different species of dinosaur converging on the one spot."
Woodside was being sought for comment.