Hill, a senior last season, announced via Twitter on Tuesday that he plans to apply for a waiver and is likely to be granted another season by the NCAA.
He missed most of the 2015 season after injuring his foot in the opener against Nebraska.
Hill, who also missed games in the 2012 and 2014 seasons because of injuries, was a Heisman Trophy candidate in 2013 after passing for 2,938 yards and rushing for 1,344 yards.
Offensive coordinator Ty Detmer said there will be an open competition at quarterback despite the presence of Tanner Mangum, who became the first BYU freshman to throw for 3,000 yards last season.
Hill is expected to be limited in spring practices as he continues to heal. Detmer said Hill has been working out and doing some light jogging but still has a way to go.
Coach Bronco Mendenhall took over at Virginia after the Las Vegas Bowl and courted Hill. They were not the only program to do so.
"He was getting recruited from all different sides, so we were part of that," Detmer said. "We talked to him several times. Coach had his wife come in, even one time. Just so everybody knew what was going on.
"We definitely wanted him here."
BYU told Hill he would get an opportunity to run a pro-style system brought in by Detmer to show NFL talent evaluators.
Detmer said there weren't any extensive conversations with Mangum about how the depth chart would shake out. He said "you never know" when asked if a two-quarterback system was a possibility.
The plan going into 2015 was to lessen the run responsibilities for Hill after the previous injuries. Detmer said he plans to do the same but added that could change.
"No. 1 is keeping the quarterback healthy, whoever that is," Detmer said. "I don't envision him running as much as he has in the past, but you never know when the game starts. He'll have some opportunities. Just him back there kind of with the threat of that, kind of keeps people at home on the back side of the run game and some of those things.
"We'll tailor our system to what the guys do best, but at the same time try to keep that quarterback healthy and out of harm's way as much as we can."
The Cougars finished the season 9-4 with impressive wins over Nebraska and Boise State. The opponents are also tough in 2016, with Arizona, Utah, UCLA, West Virginia, Michigan State, Mississippi State and Boise State on the schedule.
Hill gets the chance to be coached by Detmer, who won the Heisman in 1990 and had a 14-year NFL career.
"Taysom is a proven commodity and is a great leader for our football team," coach Kalani Sitake said in a statement. "We are thrilled to have him back for one more year."
Information from The Associated Press was used in this report.