Some might say "The Ottoman Empire" but the Ottomans were mostly Turkey and North Africa. I mean, how did what is today Iraq, Syria, Jordan, etc. naturally align before European powers started drawing lines in the sand?
Iraq, Syria, Jordan were all part of the Ottoman Empire prior to WWI.
Here is a map of the Ottoman Empire in 1914:
The region was under the Ottomans, being divides into various vilayets generally ruled by local governors.
Just to add, the French were also involved with the division, it wasn't solely a British enterprise.
It was a constantly shifting patchwork of Caliphates, Sultanates and steppe hordes with the occasional crusader kingdom for thousands of years, and before that it was many different ancient empires. You're talking about a very broad time period. At one point, much of the Middle East might've been the Persian Empire. At another, the Ottomans. At another, the Umayyads. At another, the Timurids. And during all of these times that the aforementioned groups existed, they were still surrounded by hundreds of other groups from rivaling Islamic empires to small, tribal groups.
The Ottoman Empire controlled Iraq, Syria, and Jordan. It also controlled parts of modern day Saudi Arabia, as well as Egypt. Prior to being under the control of the Ottoman Empire, Iraq/Syria/Jordan/Egypt were under the control of the Mamluk Sultanate. After the Mamluks were defeated by the Ottomans in 1517, the Ottomans controlled all of these areas (Egypt regained independence for a short time 250 years later) until the dissolution of the Ottoman Empire following defeat during World War 1 (prior to World War 1, the Ottoman Empire had been shrinking since the beginning of the 19th century, losing their holdings in Northern Africa and the Balkans).
During the period of 1501-1736, Iran (and parts of the surrounding area) was under the control of the Persian Safavid Dynasty, the Afsharid Dynasty took over starting in 1736 and continuing until 1796, and the Qajar Dynasty ruled from 1785 to 1925. These dynasties were either Persian (Safavid), or Persianized peoples of Turkic origin (Afsharid and Qajar).