There is a route known as the East Atlantic Flyway used for migration between Eastern Canada, Greenland, Iceland, Siberia, Northern Europe, Western Europe, and on to Africa.
There are 297 species using the flyway and tens of thousands of individuals per year. They are mainly waterfowl and waders, but an interesting exception is the recently discovered route of a population of Northern Wheatears in Eastern Canada - a small songbird that migrates through Europe to winter in North Africa.
There's been a lot of interest in this flyway recently as it provides a vector for Avian Flu to move between European and American populations, with many migration routes overlapping in Iceland, Greenland and Eastern Canada.
I'm not aware of any bird from south of Arctic Canada that makes it over to Europe, apart from the occasional exotic blown over by storms. Most migrations are on a north-south axis, as populations follow food sources across the seasons. There would be little advantage for a bird from lower latitudes in the Americas to make a strenuous migration to a similar latitude and season in Europe.