Fukudome was offered a one-year, $1 million deal.
Fukudome, 34, a native of Japan, finished last season with the White Sox's AL Central competitors the Cleveland Indians. He made $14.5 million last year.
Fukudome has a career batting average of .260 and a career on-base percentage of .361. He joined the Cubs in 2008 and was traded to the Indians during 2011. The Indians chose not to re-sign him for the 2012 season.
