In addition to Lars Bosteen's answer about modern migration, [several hundred thousand South American people have migrated to Spain in the last decades][1], and Brazilians have became [the largest group of foreigners in Portugal][2]. Other European countries with fewer ties and common background with America seem to host smaller populations.

Furthermore, if the "native-americans" tag in the question means that the OP is more interested in migration of descendants of native Americans than migration of Americans of European descent, the migration of Ecuadorians to Spain may qualify as the largest trans-Atlantic migration of people of native American or mixed descent.

  [1]: https://en.wikipedia.org/wiki/Immigration_to_Spain#Major_immigration
  [2]: https://en.wikipedia.org/wiki/Immigration_to_Portugal