Eaton has been out since Aug. 9 with a strained right oblique. Three days earlier he had crashed into the U.S. Cellular Field fence at full speed.
"Hopefully, he'll rejoin us sometime early next week," White Sox general manager Rick Hahn said. "We don't have specific days laid out yet, other than I don't think it's going to be Sunday in New York because of logistics and travel."
The White Sox would next open a seven-game homestand Tuesday against the Cleveland Indians.
Eaton, who also has dealt with leg injuries and an injured finger on his right hand, has been helped and hurt by his aggressive play.
"Well, we did talk earlier in the year about the headfirst slide and stuff like that, which obviously has played a role in some of his issues," Hahn said. "I think he's learned a little bit about himself in terms of, I don’t want to say toning it down, because part of what makes him good is that aggressive play. But sort of being more selectively aggressive as a means of preserving his health."
White Sox manager Robin Ventura put that selectively aggressive play in different terms.
"This is not a guy that's played 10 years, so he's got stuff that he'll learn as he goes along -- how important he is to the team, when to run face-first into the wall and when not to when the ball's 10 feet over the fence," Ventura said. "Hopefully, he can learn that."