The source characterized the team's interest in Dye, who has 325 career home runs in 14 Major League seasons, as "low on likelihood."
The name is intriguing because of Dye's power. He has averaged 31 homers over the past six seasons and hit 27 home runs last year with the Chicago White Sox to go with a .340 on-base percentage.
Dye is a career .287 hitter against left-handed pitching.
Padres general manager Jed Hoyer has maintained since taking the job in October that he wants to find a right-handed hitting outfielder, preferably a center fielder that can platoon with left-handed hitting Tony Gwynn.
Dye has been a right fielder nearly his entire career and played one season in the National League, that being his 1996 rookie season with the Atlanta Braves.
Dye, who hit .179 with seven home runs after the All-Star break, is considered a below-average defender.
His UZR/150 (Ultimate Zone Rating per 150 games) was -24.5 in 2009, the lowest of any outfielder in the American League.
The Padres play six Interleague road games, which would allow Dye to be a designated hitter, though most of his playing time would come in either left or right field.
Hoyer said Tuesday the Padres have a "bunch of offers" out to free agents. It's believed two of those have gone to veteran outfielders Jerry Hairston Jr., and Randy Winn, two players who, unlike Dye, can play center field.
Hoyer also said Tuesday that he's unwilling to offer more than a one-year contract -- with the remote possibility of one year plus an option -- to any outfielder this offseason.
Corey Brock is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.