Except that "The Levant" is more limited geographically than "Near East". "The Levant" is specifically the countries that border the Eastern Mediterranean, while "Near East" includes countries that are further east. For example, Iraq is not in the Levant, but it is in the Near East.
If you go back far enough, "Oriental"/"The Orient" was the term. The University of Chicago still has the Oriental Institute: "The Oriental Institute is a research organization and museum devoted to the study of the ancient Near East. Founded in 1919 by James Henry Breasted, the Institute, a part of the University of Chicago, is an internationally recognized pioneer in the archaeology, philology, and history of early Near Eastern civilizations."
http://oi.uchicago.edu/In 1950 James Pritchard wrote
Ancient Near Eastern Texts Relating to the Old Testament. (I know this isn't as old as Bob C.'s source in #3, but ANET is a standard work in the study of biblical archaeology and history.)