Good question!
There's been plenty of speculation on this topic in the past 50 years due to its singularity in light of the rest of the Americas. Iron? In prehistoric North America? What?
Years back, the go-to explanation was wrecked Japanese vessels. Boats from Japan do, in fact, wash up occasionally on the American Pacific coast, so it's not an unlikely proposition. Troves of iron goods at sites like Ozette Village in Washington appear salvaged rather than crafted, and some iron samples exhibit carbon and mineral signatures similar to those in Japanese iron. See retarredroof's response for why this theory still holds up.
We do also have good evidence that Pacific Northwest natives, and their northern neighbors, did in fact know how to work iron. We see iron in Siberia as early as the first millennium BC. Around 200 AD, we see the first iron tools on the Alaskan side of the Bering Strait, likely the product of Siberian trade. The Ipiutak and Thule (proto-Inuit) cultures of the first millennium AD also possessed iron tools, and by 1200 AD we see wrought iron on the Alaskan Coast. These iron implements have not as of yet been sourced to a particular ore, so we cannot confirm that the iron is indeed Siberian. But given the metal's use through the arctic and sub-arctic regions and the known interactions across the Straits, it's very probable.
The greatest testaments to the antiquity of metalworking among the Northwestern groups come from the reports of Europeans who first made contact. During his travels along southern Alaska, Captain James Cook reported that natives
never asked for either iron or copper, articles which they seemed hold in small account, and with which they appeared to be well supplied
Cook theorized that they had traded some distance for the iron. On Vancouver Island, his crew acquired local iron blades which they claimed were
certainly of their own forming and helving
In his 1792 expedition through Alaska, the Spaniard Jacinto CaamaƱo observed that locals possessed and manufactured large amounts of iron equipment, including a pretty nifty weapon:
Everyone carries a sheath knife slung around the neck [...] consisting of a blade some twelve inches long and four in width. The pommel encloses another smaller knife about six inches long [...] which is used to give the first blows and for wounding the face [...] These knives were so well fashioned and finished that at first I felt sure they were not of native manufacture, but later I found that the Indians make them themselves quite easily from the iron that they obtain by barter, heating it in the fire and forging it by beating it with stones in the water.
Locals also demonstrated obvious familiarity with iron when trading. From the first interatcions between the Spanish and native sub-arctic groups, the indigenous folks were eager to trade for iron but:
they wanted large pieces with a cutting edge, such as swords, wood-knives and the like for on being shown ribands they intimate that they were of trifiling value
Clearly the Spanish were not the first people with whom they had traded for iron.
To answer your question, we have no direct, indisputable evidence that the iron came from Asia. But we do see a long history of iron in the region with a parallel development to Siberia and colonial accounts of natives who knew how to work iron, valued it, and showed familiarity with its types and qualities. Trade with Asia is simply the best theory for this evidence.
(summarized from Steve Acheson's chapter in Emerging from the Mist: Studies in Northwest Coast Culture History, 2003, pg 214)
Ames and Maschner cite Quimby in explaining pre-contact iron working as deriving from the salvaging of shipwrecks. More recent studies showed that both iron tools and bamboo were present at the Ozette site, a clearly prehistoric site (late sixteenth, early 17 th century. Some of the shipwrecks may have been of very early Asian origin. More likely wrecks bearing iron and other metals were part of early trans-pacific trade. Grant Keddie argues that they were probably remnants of ships engaged in the Japan/Mexico Mercantile System following the mid-1500s here.
Quimby, George I. (1985). Japanese Wrecks, Iron Tools, and Prehistoric Indians of the Northwest Coast. Artic Anthropology 22(2):7-15
Edit: grammar