There's a few posts in this thread commenting on alleged links - both in DNA terms and the similarity of ancient artefacts, between native Americans and the Basque people.
Today's Quora contained an article describing some linguistic coincidences, such as the Basque word for dog (txakurra) phonetically resembling dog in the Alaskan Haida language (Xa).
A quick Google reveals other similarities between Basque and Native American vocabulary:
Basque everything "dena" ; Alaskan the people or all of us "dene"
Basque to eat "jan" ; Mattole (Native Californian) to eat "yan"; Hupa to eat "tang"
Basque: Moon "ilargia" ; Sarcee (Alberta) Moon " Yilnágha "
Even the name Alaska (from the Aleut language) is reminiscent of the Basque alatz (miracles) -ka (suffix denoting continuous action, unending).
So Alaska could mean the land of continual miracles?
There's a long list of apparent similarities given in the link below. Whilst I'm not convinced, it is interesting and would merit further research.
https://faculty.ucr.edu/~legneref/bronze/eskimo.htm#vocabularycompare