Loftin said A&M will "honor the terms" of Byrne's contract, which runs through 2013.
"What his role will be versus another A.D. coming in ... what can I say?" Loftin told the newspaper. "At some point, you have to make a transition."
A second A&M official told the newspaper the two sides were reaching an "amicable" resolution that would give Byrne a "natural transition" to another position.
As for when that transition would occur, Loftin said it would "take place at the right month and right time. For both him and A&M."
Byrne has been the athletic director at A&M for almost 10 years.