There is a book, "1421: The Year China Discovered America" by Gavin Menzies that is about a fleet of chinese ships that circumnavigated the globe before Columbus sailed his ships in 1492, 71 years later.
Then again, Leif Ericson is attributed as the first European to "discover" America circa 1000 and their "Vinland" settlement has been discovered in Newfoundland.
If you do a Yahoo Search for chinese discovered america, you'll find articles about the book from CNN and other sources.
I've purchased the book but, alas, have had no time to read it.
As for the concept of "discovery", I agree with others that the first human inhabitants are the true "discoverers" whether they related the information back to their homelands or not.
Asians are often attributed as the first settlers of the Americas and these people were the ancestors of the Native Americans.
However, there's now developing evidence the first wave of settlers of the Americas were not Asian but rather common ancestors of the Australian Aborigines or Australian Aborigines themselves. These peoples are believed to have been conquered by and interbred with the later wave of Asian settlers. Genetic testing of native Tierra del Fuegans are part of the evidence offered for this theory.