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:

The New Yorks, huh? Not sure that one would have stuck long-term.