The phrase itself was common before it was the title of Vattel’s treatise. It would be interesting to get a constitutional scholar’s take on it. I believe that there were conventions for writing titles of books, plays, poems, and so on, and I believe if they intended to reference a specific text, they would have made that very clear. They could have also used words.
I don’t believe the Constitution acknowledges any of the sources of its content, so I do not believe the Framers would have named a specific book about international law and not the many other prominent influences on their thinking. Why this one? I could probably find out if I spent the time on it, but it’s not on my “open question” list; in fact, I am 100% certain it’s not the book that is named but the concept of the law of nations, which was a term well known to them that represented a body of international law. I’d bet on it, actually, because I’m that sure.
Briefly, yes, legal treatises regarding the Law of Nations exist from many centuries throughout the history of international relations. While Vattel’s work was not the singular source utilized by the Founding Fathers, they did indeed remark on numerous occasions about using it as an important source for their legal research for the Constitution and other legal activities. Several Supreme Court decisions have acknowledged Vattel as a source for the law regarding citizenship. It takes time to assemble the citations, but they do esit. Leo Donofrio and others have cited a number of the sources for anyone who wants to take the time to consult them.