There's always room for discussion, but perhaps this previous topic will answer your inquiry.
There is evidence to say that they did or rather pre-mongol Siberian natives did and this can be found in the DNA of the Inuit, and other native American tribes down to the Rockies. Anthropologists believe that there were 2-3 migrations from the Asian continent, across the Bering Strait land bridge and into the North America. For reference :