The team has until Aug. 16 to come to terms with the young men or they will re-enter the 2011 Draft.
"I'm kind of cautiously optimistic on both of them, really," said White Sox director of amateur scouting Doug Laumann, in Chicago to finalize matters with the drop-dead date on Monday. "Both of them are satisfied with what we are trying to do for getting them done.
"We've kind of gone to the wall with both as far as what we wanted to do. It's in their hands now."
Grimes, a 6-foot-5, 18-year-old right-handed pitcher out of Millcreek High School in Georgia, has the option of playing for Georgia Tech. Terry, a 20-year-old middle infielder from Cerritos Junior College in California, could opt for collegiate ball at Long Beach State.