The two regions had deep contacts from the very beginning of recorded history. There's a huge amount to say about this subject, but I think an overview of the exchange of goods and ideas between Mesopotamia and Egypt in the 4th and 3rd millennia BCE might be a good start. After that I will talk about diplomacy and movement in the 2nd millennium BCE, and then (time permitting) the struggle for power in the 1st millennium BCE.

As early as the Naqada II period (around 3600-3350 BCE), Egypt imported goods like pottery and lapis lazuli from the Near East, and there are many examples of jugs of specifically Mesopotamian design. Cylinder seals of a similar style to those found in Late Uruk (3500-3100 BCE) and Jemdet Nasr (3100-2900 BCE) Mesopotamia have been found in Naqada II and III burials. Lead artifacts and smelted silver appeared very suddenly in Egypt during the Naqada II period, possibly the result of know-how imported from Mesopotamia or its neighbours. The earliest known writing in Egypt consists of little inventory tags which were attached to goods and writing on pottery and monuments; until recently it seemed that it was likely derived from Proto-Cuneiform tags but the picture is less clear now due to recent discoveries of tags in the Naqada III period. The technology may have come from Mesopotamia or developed independently or semi-independently in both regions. Various artistic motifs are also thought to have been shared between the two regions, like the serpopard and the priest king. Egyptian First Dynasty tombs share architectural commonalities with Mesopotamian temples, including recessed panelling on the outer walls, intricate facades with three rows of stretchers alternating with a row of headers, and short timbers used to strengthen niches in the wall. Looking at finds from Mesopotamia, figurines of monkeys at Tell Brak and the simultaneous emergence of a kind of pear-shaped mace head from Egypt to Elam have been argued to be the result of exchange in the other direction. One thing to note here is that we don't know if any Egyptian or Mesopotamian individuals were moving from one region to the other; all of this contact could have happened via Palestine and Syria.

We have a remarkable amount of information about Near Eastern diplomacy in the mid 2nd millennium due to the find of many diplomatic letters at sites such as Amarna in Egypt. There was a hierarchy of nations and particular etiquette to follow in addressing the rulers of superior, inferior, or equal states. The Great Powers Club included Kassite Babylonia, the Mitanni Empire, and New Kingdom Egypt, which also had hegemony over much of the Levantine coast. A letter (EA1) from the Pharaoh to Kadašman-Enlil I of the Kassite Babylonian dynasty begins as follows:

Say to Kadašman-Enlil, I the king of Karaduniše, my brother: Thus Nibmuarea, Great King, the king of Egypt, your brother. For me all goes well. For you may all go well. For your household, for your wives, for your sons, for your magnates, your horses, your chariots, for your countries, may all go very well. For me all goes well. For my household, for my wives, for my sons, for my magnates, my horses, the numerous troops, all goes well, and in my countries all goes very well...

When the Middle Assyrian Empire began to assert itself in the 14th century BCE, Egypt seemed to accept it into the Great Powers Club of the day with little pushback. This irritated the Babylonians, who wanted the Assyrians to be considered their subjects. Burna-Buriaš II wrote to the Pharaoh reminding him that Babylonia had previously refused the overtures of rebellious Canaanite subjects of Egypt (EA9):

In the time of Kurigalzu, my ancestor, all the Canaanites wrote here to him, saying, "Come to the border of the country so we can revolt and be allied with you." My ancestor sent them this (reply), saying, "Forget about being allied with me. If you become enemies of the king of Egypt, and are allied with anyone else, will I not then come and plunder you? How can there be an alliance with me?" For the sake of your ancestor my ancestor did not listen to them. Now, as for my Assyrian vassals, I was not the one who sent them to you. Why on their own authority have they come to your country? If you love me, they will conduct no business whatsoever. Send them off to me empty-handed. I send to you as your greeting-gift 3 minas of genuine lapis lazuli and 5 teams of horses for 5 wooden chariots.

Subjects they discussed included marriage diplomacy. Kadašman-Enlil I wrote to the Pharaoh (Amenophis III?) asking for a daughter of his to take in marriage, or even a beautiful woman who they could pretend was the Pharaoh's daughter, in order to cement an alliance. Letters also attest to the presence of Mesopotamian and Syrian craftsmen and household women in Egypt during the New Kingdom, such as one from Amenophis II to his viceroy over Kush, which includes the passage:

O you [possessor of a] woman from Babylon, a maidservant from Byblos, a young maiden from Alalakh, and an old woman from Arapshka, the people of Takhsy (in Syria) are all of no account. Of what use are they?

During the 12th century BCE, the world-system of the late bronze age collapsed. Egypt was in turmoil and lost its hold over its Levantine subjects and possessions.

In the 7th century BCE, the Neo-Assyrian Empire under Esarhaddon briefly conquered Egypt. After his death, the Nubian king Taharqa took Memphis; Esarhaddon's successor Ashurbanipal had to return to Egypt and forced Taharqa to flee to Thebes. Assyrian vassals in Lower Egypt continued to resist Assyrian rule, but after Ashurbanipal had purged them and removed Taharqa from the picture, the Assyrians had to find a way to pacify their Egyptian vassals. In order to do so, they executed all but one of the minor kings of Egypt and set Necho of Sais as their sole vassal king over all Egypt. Taharqa's son revolted against him but the Assyrians returned and forced him to flee to Nubia.

The Saite dynasty remained loose client kings to the Assyrians, who gave them a great deal of independence. When the Neo-Assyrian Empire fell apart due to the revolt by the Babylonians and their Median allies, the king of Assyria left the care of his Levantine possessions in the hands of his Egyptian vassal. The Egyptians continued sending troops to help the Assyrians, but the new Babylonian empire achieved complete victory over Assyria and set their sights on taking the Levant as well.

Conflict between these powers went on for some fifty years, though the Neo-Babylonian empire would occupy the entire region and even attempt an invasion of Egypt. The Egyptians continued to support resistance to Babylonian rule and even continued to launch campaigns that attacked Tyre and captured Cyprus. During the reign of the last Neo-Babylonian king, Nabonidus, there was little interest in campaigning in the west, as he launched only two campaigns in the Levant, neither of which seems to have drawn the Egyptians out to oppose him. The power of Achamaemenid Persia was rising in the east, and Nabonidus may have been working to arrange the balance of power between the three empires; we do have an account of a daughter of the Pharaoh sent to become a wife of Cyrus the Great.

The balance of power did not last. Cyrus invaded and rapidly conquered the Neo-Babylonian Empire, and his son Cambyses would go on to conquer Egypt.