The website Irish Sports Daily earlier reported that Kiel would transfer.
Notre Dame had no comment. A message left for Kiel's father, Kip, was not returned.
The Columbus (Ind.) High School product redshirted this past fall during the Fighting Irish's run to the Discover BCS National Championship. He is the nephew of former Notre Dame quarterback Blair Kiel, who died last Easter at the age of 50.
During his last meeting with reporters on Jan. 5, Kiel said the thought of transferring had not crossed his mind.
"It honestly hasn't," he said during media day at Sun Life Stadium. "I've been more focused on the season and been more focused on school and other things that are more important to me."
Redshirt sophomore Andrew Hendrix was ahead of Kiel on the depth chart last year as well, and Notre Dame welcomed in another quarterback this semester in early enrollee Malik Zaire.
"Anything could happen," Kiel said Jan. 5. "[Golson] could go to the NFL -- you never know. He could also get hurt, other people could get hurt -- there's so many doors that could be opened that I just have to be patient, just wait my turn, wait to get better."
"I'm gonna try, but he's a great quarterback," he added when asked about unseating Golson this spring. "He's going to lead this team and he's going to keep getting better, so I just need to keep getting better and learn from him, and also learn from coach [Brian] Kelly and just try to keep getting better each year."
After spurning LSU at the 11th hour last January to enroll early at Notre Dame, Kiel drew sharp criticism from LSU coach Les Miles, who questioned the heart of Kiel at the Tiger Gridiron Club's annual national signing day "Bayou Bash" in Baton Rouge, La.
"There was a gentleman from Indiana that thought about coming to the Bayou state," Miles said. "He did not necessarily have the chest and the ability to lead a program, so you know."
Kiel took the high road whenever asked about Miles.
"Once he said it, I don't blame him at all -- I was dumb during the recruiting process," Kiel said Jan. 5. "I couldn't really make up my mind, but I was also doing what was best for me at that time, and that was picking a college. That much pressure and being so highly ranked, I couldn't make up my mind.
"It was hard, but I'm very happy with the decision I made, and I've always respected Coach Miles. I'm not going to throw him under the bus, because I think he's a great coach. Everything he stands for, I think, is great. He's a winning coach. There's nothing more to say. I can't take back what I did, so I respect Coach Miles and everything he stands for."