Looking to become the seventh and youngest member of baseball's 600-homer club, Rodriguez has gone 21 at-bats since slugging No. 599 last Thursday against the Royals, including taking an 0-for-4 on Tuesday against the Indians.
"It'd be a good time to do it," Yankees manager Joe Girardi said before the game. "It's going to happen. It's just a matter of when."
Rodriguez has homered five times on his birthday, doing so in 1996, 2001, twice in '02 and in '04.
According to the Elias Sports Bureau, only two players have hit a century milestone home run on their birthdays: Cecil Fielder hit No. 300 on Sept. 21, 1997, and Raul Ibanez hit No. 200 on June 2, 2009.
Rodriguez's first hacks of Tuesday came against rookie right-hander Josh Tomlin, who made his Major League debut for the Indians.