Chapman sustained fractures over his left eye and nose and had a titanium plate inserted to stabilize the area on Thursday night at a Phoenix hospital.
Perez visited Chapman at the hospital just after the accident ended the Royals-Reds game in the sixth inning on Wednesday night.
Reds medical director Dr. Timothy Kremchek estimated that Chapman could resume working out and throwing within 10 to 14 days and pitch in game conditions in four to six weeks.
Dick Kaegel is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.