Yes.

Thanks to the work of the Alaska Native Language Center in Fairbanks (and specifically Edward Vajda of Western Washington University), we have a very good grasp of linguistic links between Siberian and Alaskan languages. The Dene-Yeniseian Hypothesis proposes a genetic relationship between the Na-Dene languages of North America and the Yeniseian languages of Siberia.

This is big stuff if you're studying the human flow across Beringia. We have archaeological and genetic evidence for the settlement of North America, and the linguistic link is something else we can point at to say that yes, North America was settled across Beringia and not some other route.

The Dene-Yniseian Hypothesis is largely based on the Ket language, spoken by a handful of hunters along the Yenisei River. Thanks to their isolation, their language has been preserved despite the ebb and flow of pastoral, horse-riding civilizations in central and northern Asia. That makes a place like Kellog, population 342, the perfect place to study.

Given the current evidence at hand, the first groups of homo sapiens sapiens to cross across Beringia came a minimum of 14 000 years ago (although most agree that this is an underestimation). To paint the picture a little more clearly, modern-looking and -behaving homo sapiens sapiens 'attained' the ability for language anywhere from 200- to 150 000 years ago, living in the Cradle of Life in Eastern Africa, given the genetic development of the FOX-P2 (or 'Language'-) gene. The accepted theory is that humans spoke languages of a similar sophistication and complexity to those which we speak today and, given the lack of contemporary textual evidence, we are unable to map out the language families that developed therein. Roughly 65- to 55 000 years ago, mass migration waves of homo sapiens sapiens hunter-gatherer groups left Eastern Africa and spread throughout Eurasia (due to the last Glacial Period lasting all throughout this era, the face of the Earth looked quite different than it does today), with their separation from the homeland leading to a number of linguistic peculiarities and the formation of thousand, if not tens of thousands, of small and large language families. With regards to future Amerindian settlers, these groups migrated through Asia at a time period best described as an explosion of population and migrations all across the planet due to the recession of glacial masses and pressure from expanding sedentary cultures in Central Eurasia.

To answer this question, it is important to understand that these groups came at very different times and that the 'Indigenous Siberians' like we have today, had not yet settled in the Siberian Tundra as it was uninhabitable at this time given the residual glacial make-up of the territory. As Stanford professor Matin W. Lewis describes: "there are at least three major language families in the Americas including Inuit-Aleut (Greenland, Northern Québec, the Canadian Territories, North- and West-Alaska and Eastern Siberia), Na-Dene (most other territories in North America; including Athabascan languages spoken in Central Alaska; possible connections to Sino-Tibetan languages in South-Western China) and the third is a family comprising the remaining languages throughout Southern North America, Central America and South America; although this is greatly contested and it is proposed that upwards of twelve families can exist". Each of these families represents a separate mass-migration across Beringia and the further South we travel, generally speaking, the earlier the population's ancestors would have crossed the Straight (in Chile, there are settlements from ca. 18 000 BCE, the earliest in the Americas).

As shown in the quote above, and to answer the question more directly: there are copious amounts of linguistic data that support and prove that there ARE linguistic ties between certain Indigenous Siberian language groups and certain Native American groups (namely the Inuit-Aleut family in the North American Tundra Zones, and perhaps [although this is far from certain] particular branches of the Na-Dene languages).

I trust this helps.