Other answers here are discussing the possibility of direct invasion or territorial intervention in the US by the major European powers, but there is another important dynamic here to consider.

During the first few decades of its existence the US was not a major power in the Atlantic, and it existed in a volatile "balance of power" international system which stressed direct competition. The US, in other words, was a small fish in a big pond. More accurately, before the Constitution the US was several small fishes in a big pond, which is where a lot of the perceived danger surfaced.

It was feared that through political and economic pressure, the major European powers might drive wedges between the various states for their own gain. If the US could be split into two or three smaller confederacies, they could be pitted against one another in ways that benefited European powers (through trade, primarily) at the expense of the states.

Many historians today like to stress that the formation of the Constitution reflected international concerns as much as domestic ones. Strengthening the national government and then funneling foreign policy and diplomatic relations through Washington was in part a way to keep side deals and backroom agreements with Europe from driving the country apart.

In the years immediately following the formation of the United States, the British would definitely have been the European Power most likely to successfully attack the United States. The Spanish empire was in decline by that time, and the French were busy either fighting each other or trying to take over Europe. Honestly, the Native Americans would probably have been viewed as the biggest threat by many Americans. While Native Americans wouldn't have been a real threat to the well established east coast cities or the US government as a whole, this country was rapidly expanding west and the new settlers would have been in constant danger of an attack by Native Americans.

At the time, Britain was the only power that could project a real invasion of a larger sovereign nation (such as the USA) on the western hemisphere.

France might also have been able, if the British did not intervene, and it is perhaps within the realm of possibility that Spain could have mustered an invasion as well.

Other powers that had a large navy at the time were the Ottomans, the Dutch and the Russians, all of which had no interests northeastern America.

An analysation of the threats as they would have been percieved by Americans (who would be a third rate power econonomically and a fifth or sixth rate power militarily at the time).

Spain.

While Spain and her Empire had been in a slow decline since the 1600s, it was still huge. See this map. However, that size also meant that there were huge swathes of empty territory. Spain had enough problems of keeping and trying to colonise the territory they already had in America without going to war to acquire more. In north America, their economical interests in America revolved mainly around red dye from lice in Mexico, with some secondary cotton, sugar and tobacco production, the latter two mainly on their islands in the Caribbean.

Spain had been an ally of the US in the war of independence and the combined Franco-Spanish armada had actually been considered a threat the the home islands by the British.

Spain at this time can be considered a second rate power, both economically and militarily.

Relations, however, were good, and the Spanish had no real interests in designs against the US, so I doubt the US government saw Spain as a threat. By the end of the Napoleonic war, the Spanish Empire in America had desintegrated, and they were even less of a threat to the US.

France.

France had some designs in America, mainly to keep and regain lost territory in the Caribbean. While France was one of the (if not the) world's foremost powers during the era in discussion, its ability to project power across the sea was severly limited by the British Royal Navy. And while France was a first rate power both economically and militarily, its interests in northeastern America laid mostly in its sugar islands (very profitable) in the Caribbean and perhaps Quebec and New Orleans. All were, however, under the threat of British occupation in case of war. There's a reason Napoleon sold the Louisiana territory (which was formally Spanish at the time) to the US - he had no way to keep it, and it brought him nothing in money or resources.

For most of the era, France and USA held aimcable relations sprung out of the alliance during the war of independence and during the Napoleonic war, France lacked the ability to punch through the Royal Navy and act on the western hemisphere.

While large political changes could have changed the willingness and ability of France and Spain to be enemies of the US, at the time, they were not and were not interested in being so.