The Red Sox are second in the AL East, and designated the 33-year-old Kotsay for assignment last week. The White Sox are second in the AL Central and sent the 27-year-old Anderson to Triple-A last week.
Kotsay hit .257 with one home run and five RBIs in 27 games for the Red Sox this year. He started out the season on the disabled list while recovering from back surgery.
"He's going to help us all the way around the field," White Sox manager Ozzie Guillen said. "It depends on who needs the rest and where we need him the most."
Anderson was hitting .238 with two homers and 13 RBIs when the White Sox sent him down to Triple-A Charlotte. The Red Sox will option him to Triple-A Pawtucket.
"He's very athletic. He's very gifted defensively," Red Sox manager Terry Francona said.
Anderson is a career .225 hitter with 20 homers in 334 games since making his major league debut with Chicago in 2005. Like Kotsay, Anderson is known for his excellent defense.
"I think that's the best, to me, that could happen to that kid. We gave him a lot of opportunities, and he had a lot of friends here," Guillen said. "In the meanwhile, we have to make the team better. I think to have him get another shot with another organization is good for both sides."