That's a pretty complex question, especially given that Americans at the time might not even identify themselves as citizens of the united states, but rather as citizens of their local state (Virginia, New York, etc). Everything below here therefore speaks in generalities. It's also complex in that it refers to Napoleon the person, as opposed to France.

In general, the northern states were much more oriented around mercantilism. Therefore, they tended to be pro-British, more then the southern states, which tended to be pro-French. Into this mix was the fact that the early french republic was very much inspired by the American revolution, and several key players in the American revolution - Lafayette and Thomas Jefferson - where also participants in the french (which much less success, and more then one close shave).

When the French revolution went bad (the reign of terror and the De-christianization of France in particular) popular opinion quickly turned on the French Revolution. The United States declined to continue to repay the French government for it's support in the revolutionary war (noting that the payments were for royalist France, not the Republic). This led France and the United States into a brief, undeclared war with France called the Quasi-War, which Napoleon ended. That, plus the Louisiana purchase led to Napoleon's popularity as the US Continued to distance herself from the demise of her "Sister Republic."

For the next decade the US, the British and the French constantly played off of each other, going through periods of near war and reconciliation. As Napoleon dominated Europe, American leaders worried about his domination of Europe and it's implications for the states. On the other hand the southern states worried more about the British.

Jefferson reflected this after the war of 1812 stating:

"Although we neither expected, nor wished any act of friendship from Bonaparte, and always detested him as a tyrant, yet he gave employment to much of the force of the nation who was our common enemy. So far, his downfall was illy timed for us; it gave to England an opportunity to turn full-handed on us, when we were unprepared. No matter, we can beat her on our own soil, leaving the laws of the ocean to be settled by the maritime powers of Europe, who are equally oppressed and insulted by the usurpations of England on that element.—To W. H. Crawford. vi, 418. Ford ed., ix, 502. (M., Feb. 1815.)"

In the end the British did not appreciate the growing US maritime presence (both commercial and militarily) and ended up pushing conscription and commercial restrictions too far, which in turn led to the war of 1812. During that war, the US made no common cause with Napoleon. Even a common enemy was not enough to pull America to Napoleon's side.

John Adams seems to summarize it fairly well when he said:

"but I hope that reason will never again rashly and hastily create such Creatures as him. Liberty, Equality, Fraternity, and Humanity will never again, I hope, blindly surrender themselves to an unbounded Ambition for national Conquests, nor implicitly commit themselves to the custody and Guardianship of Arms and Heroes. If they do, they will again end in St. Helena."