Yes, this would be perceived as a strange request and will very likely be turned down, and may even be frowned upon by certain people. You have identified a strange blind spot in the U.S. academic employment system -- U.S. universities do not really have a good understanding of some of the issues facing foreign scholars. I also think this ties in to cultural factors such as the strong U.S. belief in self-reliance and personal responsibility, and a negative bias towards people of modest means. Asking one's employer for an advance is generally seen as a somewhat negative thing that only very poor or desperate people would do.
Let me offer a personal anecdote that could add some context. Some years ago I had a similar experience to you when I arrived in the U.S. as a postdoc. Like you, in my first few weeks I had to spend a nontrivial amount of money on a rent deposit and other necessary expenses, all with no financial assistance from my university. Things got worse when my first paycheck arrived and I discovered that they had entered my salary incorrectly in the system so the amount was lower than I expected. I asked for this to be corrected and was told that it will be and that the change will be reflected in the next paycheck, one month hence. Another upsetting thing was that I was asked by my department to pay $50 as a deposit for my office key, a key to the computer lab and a building entry card. As it happened, I had enough savings so all of this did not pose a great difficulty for me, but I knew this might not be true of all postdocs, and was very annoyed by the principle that I had traveled from halfway around the world to contribute my work and talent to the university and was being treated with such thoughtlessness and lack of consideration, to the extent that instead of them paying me for my efforts we were starting off with the money flowing in the opposite direction!
When I mentioned these complaints to the professor I was working with, he immediately offered to give me a loan (which as I said was not necessary in my case). He then chuckled and recalled with nostalgia how when he was a young postdoc, moving to the U.S. from another country as a postdoc, he found himself short of cash for precisely the same reason. His own research mentor happily gave him a loan to tide him over until he started getting paid. So, he understood the problem very well. However, he did not appear concerned by this approach and thought it was very natural that this cycle of personal loans should continue as a solution to the problem...
I should also add that part of the reason for my annoyance was a completely opposite experiece I had a couple of years before, when I arrived in France to work as a postdoc on a French government grant. On the day of my arrival I was given an envelope with 500 French francs in cash, and had prearranged housing with rent being deducted directly from my paycheck and no deposit necessary. Needless to say I appreciated very much this thoughtful attitude on the part of my French hosts. Life in the U.S. has many wonderful aspects to it, as you'll soon find out, but on this particular issue the comparison to France is not especially flattering.