The Inuit in Greenland have a considerably about of folklore (some of it history, some of it historical fiction) concerning their encounter with the Norse. I discussed some of the most well known ones in this post. The people of Ammassalik also have a story about five brothers getting successively lost at sea (each brother going on out in search of previous one to get lost) and eventually finding each other again in Akilineq, a populated land to the east of Greenland. "Akilineq" is a common Inuit place-name, and means "the opposite shore" shore. So whether we're to assume that these brothers made it to Iceland or to some generic legendary land across the sea is debatable.

The most famous legendary pre-Columbian explorer in the Americas is Topa Inca Yupanqui, grandfather of Atahualpa (the emperor when Pizarro arrived). The earliest surviving account of Topa Inca Yupanqui's voyage was written down in Pedro Sarmiento de Gamboa's History of the Incas in 1572, about 90 years after the voyage took place. Topa Inca Yupanqui had just finished conquering the Gulf of Guayaquil when traders arrived in Tumbez (modern Tumbes, Peru) from the west. Inspired by the description these traders gave of their homeland, Topa Inca Yupanqui ordered that a large fleet of balsas be constructed in order to carry an army of 20,000 men. After 9-12 months, the fleet returned with captives and other trophies of their expedition after discovering the islands of Ninachumbi and Avachumbi. The trophies were stored in Saksaywaman, the fortress near Cuzco, and the caretaker of these items, Urco Huaranca, was Sarmiento de Gamboa's main source for this information.

Now the Gulf of Guayaquil had been a major center for Pre-columbian maritime trade, and the people there had been sailing between Ecuador and Central America / southern Mexico for centuries, so having recently conquered the area, Topa Inca Yapanqui would have had the ability to undertake such an expedition. But merely having the ability to do something doesn't confirm that he actually did. There's a lot of unanswered questions here. Were the traders in this account actually from the north rather than the west? Were they Polynesians from the west (see this post for information on Pre-Columbian South American-Polynesian contact)? Did these traders even show up at all? Where are Ninachumbi and Avachumbi? They're generally associated with Galapagos Islands, but that wouldn't explain the artifacts that were brought back. Among these were said to be a brass chair, and the skin and jaw of a horse. Sarmiento de Gamboa doesn't mention whether he saw these items for himself or whether Urco Huaranca only told him about the items under his care. A horse jaw and skin, assuming they were correctly identified, casts considerable doubt on the story as a whole. Certainly no such animals lived on the Galapagos (nor people, for that matter, to be taken captive). Topa Inca Yapanqui would have had to sail all the way to Asia and back in less than a year to bring back the remains of a horse prior to the arrival of Europeans in the Americas. If these items really existed, then perhaps they were misidentified / mistranslated by Sarmiento de Gamboa, but without knowing what they might really have been they don't help us much now. So ultimately, we're left with a lot of hypotheticals that can't be confirmed now.