I was doing research once, and found this quote, I've wondered about it a lot. Did it apply to all tribes, or just some. If it is just some, then would that mean that people have come to the Americas multiple times. It seems pretty significant, but I haven't heard about it, so I don't know if it's a myth.
The Tartar Chinese speak the dialect of the Apaches. The Apaches bear a striking resemblance to the Tartar. About the year 1885, W. B. Horton, who had served as County Superintendent of Schools, at Tucson, was appointed Post Trader at Camp Apache, and went to San Francisco to purchase his stock, where he hired a Chinese cook. His kitchen adjoined his sleeping apartment, and one evening while in his room he heard in the kitchen some Indians talking. Wondering what they were doing there at that hour of the night, he opened the door and found his cook conversing with an Apache. He asked his cook where he had acquired the Indian language. The cook said: "He speak all same me. I Tartar Chinese; he speak same me, little different, not much." At Williams, in Navajo County, is another Tartar Chinaman, Gee Jim, who converses freely with the Apaches in his native language. From these facts it would seem that the Apache is of Tartar origin.
From the fact that the Apache language was practically the same as that of the Tartar Chinese, color is given to the theory advanced by Bancroft in his "Native Races," Volume 5, p. 33, et seq., that Western America was "originally peopled by the Chinese, or, at least, that the greater part of the new world civilization may be attributed to these people.''
https://books.google.com/books?id=JUUOAAAAIAAJ&dq=history%20of%20arizona&pg=PA13#v=onepage&q&f=false
Speaking as someone with a little amateur linguistic background, this source sounds like a bunch of nonsense to me. While is is popularly hypothesized that some New World languages (including Navajo and Apache languages) are related to Siberian languages, they are in no way mutually intelligible. This proposed language family is called the Dené–Yeniseian language family, and while still unconfirmed, is regarded in some linguistic circles as plausible. It's bizarre that your source seems to hint at this idea, given that the it dates to the early 20th century, and the Dené–Yeniseian hypothesis is much newer, the oldest paper written on the matter comes from 1998 (1). On top of that, Chinese "Tartars", or more appropriately, Tatars, speak a Turkic language, not a Dené–Yeniseian language.
Even assuming that the author made a mistake in referring to the people as Tatar, the Old World speakers of Dené–Yeniseian languages are tiny in number, most languages in the Yeniseian family went extinct by the mid 19th century, and today, Ket is the only extant language, with maybe 200 speakers (2). Their tiny numbers, and their position in the center of Siberia meant that almost certainly no Yeniseian speakers immigrated to the US.
And again, while they may be related, they are not at all mutually intelligible with each other save for some passing similarities (1).
The Apache have six languages, all which are classified under Athabaskan/Dené language family. There are over forty Athabaskan languages used by indigenous peoples in the American Southwest, the Pacific North coast, and throughout Alaska and Northwestern Canada. As far as I am aware, there is no current scholarship that links these languages to Chinese.
This question was also covered in r/linguistics not too long ago - https://np.reddit.com/r/linguistics/comments/4e1hpm/any_other_source_that_shows_apache_and_tartar/
This relation between the Tatar and any Apache language has no merit whatsoever. To cite Hubert Howe Bancroft in the work referred to above, The native races of the Pacific states of North America (1874)
The American Indians, their origin and consanguinity, have, from the days of Columbus to the present time proved no less a knotty question. Schoolmen and scientists count their theories by hundreds, each sustaining some pet conjecture, with a logical clearness equaled only by the facility with which he demolishes all the rest. One proves their origin by holy writ; another by the writing of ancient philosophers; another by the sage sayings of the Fathers. One discovers in them Phoenician merchants; another the ten lost tribes of Israel. They are tracked with equal certainty from Scandinavia, from Ireland, from Iceland, from Greenland, across Bering Strait, across the northern Pacific, the southern Pacific, from the Polynesian Islands, from Australia, from Africa. Venturesome Carthaginian were thrown upon the eastern shore; Japanese junks on the western.
Bancroft later also mentions proposed Chinese origins and concludes the paragraph with:
[They are] showing the different fancies of different men and times. Fancies, I say, for modern scholars, with the aid of all the new revelations of science, do not appear in their investigations to arrive one whit nearer an indubitable conclusion.
From a linguistic standpoint, there is nothing, that suggest a relationship between Mandarin or Tatar and Apachean.
The closest you will get to this, is the proposed language family of Dené-Yeniseian. It connects Na-Dené languages, including Southern Athabaskan (Apachean), with the Yeniseian languages around the Yenisei river in Russian Siberia. The only language of the latter family that is not extinct is Ket, which has only 200 speakers left. Edward Vajda is the main proponent of this theory. The connection between the two families, however, is much too disputed to provide any insight into historical events.
As your question was also about New and Old World contact, let me tell you about the Yupik languages. They belong to the Eskimo-Aleut language family, and are spoken in Asia as well as North America. This does, however, not point to any pre-columbian contact between civilizations, but is the result of migration. As you might know, the Bering strait was frozen 20,000 to 8,000 years ago. This allowed humans to migrate into the Americas from Asia, bringing their language with them. This is the reason of the existence of related languages and peoples in Siberia and Alaska. This would also be the most likely explanation for the existence of the Dené-Yeniseian language family on different continents. Genetic relations in languages do not point to 'modern' contact.
What does point to contact in languages, though, are loanwords. Proto-Polynesian *kumala 'sweet potato' might be borrowed from South America, for example Quechua k'umar 'sweet potato', together with the sweet potato's introduction into the Polynesian world. It has to be noted, though, that the agricultural contact could have very well happened by accident, lost cargo of South American origin reaching Polynesian islands, and the similarities of the words could just as well be coincidental.
In general, this is a prime example of why historians prefer recent sources to more distant ones. This source is 98 years old, and I can find no evidence that Thomas Edwin Farish was trained in either historical or linguistic research, at least not as we understand them today. This book seems like an excellent source for understanding the culture and mindset of a subset of white Arizonans nostalgic for the "Old West" in and around 1915, but it's not reliable as a source of historical facts, no matter how stridently Farish asserts that he is reporting only fact.
It would be one thing if he made such a claim, and then that claim was tested and ratified by several subsequent generations of historians, linguists, and historical linguists, but it isn't. Just one claim, made one hundred years ago, does not good history make.
I'm not trying to admonish you, of course. But this is a great object lesson in the nature of historical sources.