After years with the WAC and then Mountain West, BYU football struck out on their own and went independent after the 2010 season.
That doesn't mean the Cougars aren't interested in a conference partnership again.
Outside of Notre Dame, BYU is probably the most competitive FBS independent out there, especially now that Navy is in the AAC for football. If a Power Five league came calling, though, the Cougars would listen.
The school had a dalliance with the Big 12 a few years ago, as one of the primary programs mentioned as potential expansion candidates. Ultimately, the league decided to stay at 10 teams.
BYU Cougars athletic director Tom Holmoe won't rule out joining a league, but it won't come for a few years.
He spoke to KUTV about the possibility of moving to a new league. BYU definitely sounds interested, but the timetable is put off until the leagues start the next round of media rights negotiations.
Holmoe put the year that it could happen at 2023. Via CollegeFootballTalk:
It sounds as though that will be the case for at least the next four or five years, as athletic director Tom Holmoe told KUTV that he doesn’t foresee realignment involving the school happening again until at least 2023 and only after then could the program consider a move to a league even if it’s not of the Power Five variety.
“When they say 2023, it’s a little bit before and a little bit after when those big conferences renew or redo their broadcast rights agreement. After that time, if there is a change for BYU, then he would have to think about it and what the future would hold for sure,” said Holmoe. “We’re paying attention to see how those things go but you’re not going to see any movement for the next couple of years.”
In terms of the current Power Five, the Big 12 probably makes the most sense for BYU. It is anyone's guess what the college football world will look like in a few years though.