The Cubs are trying to bolster their bullpen and have had discussions with the Braves and Nationals about the outfielder.
A Major League source said Chicago offered Byrd, who's in the final year of a contract that pays $6.5 million in 2012, to the Nationals for pitching.
Another source said the Cubs don't appear to be shopping Byrd as much as listening to offers.
Cubs Discussing Byrd With Braves, Nationals