Before they became the New York Yankees, they were the New York Highlanders from 1903-12. A few months prior to their first official season as the Yankees -- "Yankees" was already popular nickname during their time as the Highlanders -- team co-owner Frank J. Farrell considered calling the team the "New Yorks" at the behest of manager Frank Chance.
Here's a New York Times article dated Jan. 16, 1913:
January 1913-Yankees co-owner Frank J. Farrell wants his team to be called "New Yorks" and not "Yankees" pic.twitter.com/UPVtQIThmY— Todd Radom (@ToddRadom) January 28, 2014
The New Yorks, huh? Not sure that one would have stuck long-term.