Author Gavin Menzies makes the claim in his new book, Who Discovered America? Among Menzies' other claims are that the first inhabitants of the Western Hemisphere didn’t come over land from the Bering Strait, but instead were Chinese sailors who first crossed the Pacific Ocean 40,000 years ago.
He also writes that DNA markers prove American Indians and other natives are the descendants of several waves of Asian settlers.
You can read this interesting story in an article by Michael Zennie published in the MailOnline at http://goo.gl/4JNqeI. However, be aware that many historians believe that Menzies' claims are pure rubbish. They have been described as historical fairy tales.
I am not an historian but I can point out that neither Christoper Columbus nor Admiral Zheng He were the first non-natives to arrive in North America. It appears that numerous Europeans had already made the trip across the Atlantic years earlier. Leif Ericson is generally recognized as the first documented arrival in North America about the year 999 A.D. However, Ericson never claimed that he made the discovery. Instead, he acknowledged that he had heard the story of lands in the west from merchant Bjarni Herjólfsson who claimed to have sighted land to the west of Greenland after having been blown off course.
Many other stories claim that even earlier European travelers crossed the Atlantic to North America but proof is lacking.
Chinese mariners certainly did possess ships and technology to cross the Pacific, but there appears to be no proof of such crossings until the claim of Chinese Admiral Zheng. His travels were made around 1418. Of course, that "proof" is somewhat less than iron-clad.