The Middle East is a term used in modernity. In ancient history the same region is called the Ancient Near East.

The British never used the term Close East, but before World War II they referred to what we now call the Middle East as the Near East. Basically, the Ottoman Empire, Persia and the Arabian peninsula was the Near East. The Middle East was Pakistan, India, Sri Lanka, Burma and Bangladesh. The Far East was any part of Asia with a Pacific coast line.
After World War II, the media in the United States began to call the Near East the Middle East. After 1947, the BBC followed suit. The term Near East has become practically extinct, although it is sometimes used in stories about Morocco, Algeria and Tunisia.

The closest term actually in use that I can think of is the 'Near East', which either refers to Anatolia or Eastern Europe, or both.