There could be a few different ways to go about studying this topic and they don’t all include a traditional master’s degree in history. You might consider interdisciplinary programs like the University of Pennsylvania’s terminal MA in Mesopotamian Studies. Just from Google, I can see that there are multiple programs out there like this.

Alternatively, can I ask why an MA and not a PhD? From my viewpoint, this isn’t a field of study that it would make a lot of sense to pursue an MA unless you have a very specific use for this as a terminal degree. (For example, you already have another degree like an MS/LIS and having the second degree in a specific area will help you land a difficult to get job like a subject librarian at a research institution.) If you want to study this field with a plan to get a job as a professor or doing other sorts of research at a museum or similar thing, you may be better served by pursuing a PhD. And even then, academia is a mess right now and I’m not even sure how likely you’d be to adjunct with a PhD in Mesopotamian History.

Regardless, I would encourage you to only pursue an MA or PhD in Mesopotamian History because you really, really love it. There aren’t a ton of jobs. A master’s will cost you money. A PhD shouldn’t cost you money but you’ll live on a tight budget unless you have some other form of income like generational wealth and in that case, go you! I don’t mean any of this to be discouraging. I just want to provide a balanced viewpoint.

Anyway, the long and short of my response is: What are you aiming to do with this degree? Your answer to that question will help narrow down exactly what programs and schools to pursue.