The move is a surprise because Flores is another right-handed bat on the bench and he will be the team's third catcher.
Flores, 26, started 2009 as the team's everyday catcher but a string of injuries has kept him from returning to the big leagues since then.
Flores is a career .260/.313/.406 hitter at the big league level.
The team also has a top prospect in Derek Norris, who the team hopes is its long-term answer at the position, making Rodriguez and possible Flores expendable. If Flores shows he's back to where he was in 2009 before his injuries (.301/.371/.505 in 29 games), either he or Ramos could bring back more in trade.
Rodriguez, 39, is a free agent after the season.For more baseball news, rumors and analysis, follow @cbssportsmlb on Twitter or subscribe to the RSS feed.