When Columbus "discovered" the New World, the Old World had little to no knowledge of it. What little was known was perhaps Greenland, and - based on accurate knowledge of the size of the world - a vast expanse of unknown between Europe and East Asia, which was assumed to be empty ocean until Columbus proved otherwise, setting off the Age of Discovery and the associated mania with finding unknown lands, some of which proved true (Australia, Antarctica).
At this time, what did the New World know of, or speculate about the Old World? They were reasonably advanced, for example the Mayans were very good at Astronomy, and so may have known the size of the world. If so, they would have realised they occupied a narrow strip of land surrounded by a vast, unknown expanse. Did they know this, and if so what did they speculate would be there?