Was Thomas Jefferson Part Middle Eastern?

Yes, according to Professor Mark Jobling of University of Leicester:

Thomas Jefferson’s haplogroup – shared with the two men from Britain
- is known as K2.

K2 makes up about 7% of the Y chromosome types found in Somalia, Oman, Egypt and Iraq. It has now been found at low frequencies in France, Spain, Portugal and Britain.

Of the K2s looked at by the study, Jefferson’s Y chromosome was most similar to that of a man from Egypt. But genetic relationships between different K2s are poorly understood, and this may have little significance.

Link