The pilots' names and countries of origin were not immediately released.
It was the second fatal crash involving a T-38C this year. Two pilots died eight days ago at Columbus Air Force Base, Miss.
Investigators are trying to determine what caused both crashes.
"At this point we have no indication that there was any tie between the two," said Capt. John Severns, Chief of Media Relations for Air Education Training Command at Randolph Air Force Base near San Antonio.
Thursday's accident was at least the sixth fatal plane crash involving Sheppard base personnel in 10 years, and the third involving a T-38C Talon trainer jets since 1995.
Seven T-38 Talons have crashed since 1997, resulting in four deaths, according to the Air Force Safety Center.
The Talon, primarily used by the Air Education Training Command, was first deployed in 1959, according to an Air Force Fact Sheet.
"We have an extensive training program," Severns said. "The T-38 is just a part of that program... The T-38 happens to be the trainer we use for those in the fighter training pipeline."
Pilots from 13 NATO countries train through Sheppard's Euro-NATO Joint Jet Pilot Training Program.
The T-38's crew normally consists of a student pilot and instructor seated in tandem. Student pilots pick up aerobatics, formation flying, night and instrument flying, cross-country awareness, general maintenance awareness and supersonic techniques in the aircraft.
All Thursday flights at Wichita Falls' Municipal Airport, which shares runways with Sheppard, were canceled, a city official told the Wichita Falls Times Record News.
Wichita Falls is about 135 miles northwest of Dallas.