The two teams already have paired up for one big swap. The White Sox sent starting pitcher Jon Garland to the Angels for shortstop Orlando Cabrera in November. Chicago has pitching needs, notably in the bullpen, and is seeking a leadoff man.
Angels manager Mike Scioscia, who has addressed the club's need for improved slugging percentage, has a history with Konerko dating to their days together in the Dodgers organization.
Reagins said he would neither confirm nor deny the talks, consistent with his policy of not commenting on rumors.
The White Sox recently acquired outfielder Nick Swisher from the A's, and he could be moved to first if Konerko is traded.
Konerko, who began his career with the Dodgers in 1997, spent 1998 with the Reds and Dodgers before getting shipped to Chicago for Mike Cameron.
Konerko has averaged 29.9 homers and 96.2 RBIs in his nine seasons with the White Sox. He's a .281 career hitter, slugging .495 with a .353 on-base percentage.
While his acquisition would satisfy the club's long pursuit for another big weapon to place behind Vladimir Guerrero, the accompanying loss of Kendrick or Figgins would remove another valuable dimension from the offense.
One factor that could weigh into the Angels' decision making is the significant difference in home ballparks. U.S. Cellular Field has been highly favorable to power hitters, while Angel Stadium is viewed as more of a pitcher-friendly park.
Of Konerko's 269 home runs in his nine seasons with the White Sox, 157 have been hit in Chicago -- more than 58 percent. When he hit 41 and 40 homers back to back in 2004 and 2005, 52 were hit at home, 29 on the road.
The Angels attempted to land Konerko as a free agent after the 2005 season, but he returned to the White Sox for five years and $60 million.
With their signing of free-agent center fielder Torii Hunter, the Angels have added a big right-handed bat to their offense as well as a seven-time Gold Glove winner.
Hunter delivered 28 homers and 107 RBIs for the Twins in 2007, batting .287. His 303 total bases exceeded the 269 total bases by Konerko, who batted .259 with 31 homers and 90 RBIs.