Konerko was diagnosed with a concussion on Wednesday and put on Major League Baseball's designated seven-day concussion list Friday.
There is no medical treatment for a concussion other than rest and lightness control.
As for Konerko's condition, the team said Sunday there is a slight improvement from when he first sustained the injury after he caught an elbow in the head on a play at first base Tuesday night.
"He has a little of everything," manager Robin Ventura said. "It is not just headaches; he doesn't look right and doesn't move around too well. He said it has gotten better over the past few days."
Konerko will have a complete concussion examination at Rush Hospital on Monday. If it is determined he is on the road to a full recovery, he will be allowed to start baseball activities on Tuesday.
According to the White Sox, the plan would then be for the 35-year-old veteran to join the team in Kansas City on Friday.
Konerko has been able to drive his own car from home to the ballpark each day since the injury occurred.