The Axis powers did have some plans to at least attack the United States (though they mostly involved long-range bombers and "fire balloons"). During the war, neither side had any serious invasion ambitions, because they were a) already pretty busy and b) oceans.

However, it doesn't take much historical speculation to suppose that, had the Axis won the war in Europe and Asia, they would have eventually turned their attention to North and South America.

But that would be speculation. What we do have is this this map.

Yup. South America, partitioned into Nazi states.

Roosevelt saw this map and said, in a speech:

"Hitler has often protested that his plans for conquest do not extend across the Atlantic Ocean ... I have in my possession a secret map, made in Germany by Hitler’s government -- by the planners of the new world order. It is a map of South America and a part of Central America as Hitler proposes to reorganize it."

Thing is, that map was almost certainly a British forgery and Roosevelt (probably) knew it. But it served his purposes. Here's another Roosevelt quote. It's not related to the map directly, but it does show where his thinking was at:

"You know, I'm a juggler, and I never let my right hand know what my left hand does. I am perfectly willing to mislead and tell untruths if it will help win the war."

Roosevelt surely engaged in some mental "what-if" about a possible invasion, but he probably had a good idea of how difficult it would be for the Nazis (or the Japanese) to attempt one, especially while simultaneously taking on most of Europe and Asia.