According to a report by The Advocate, LSU did enter into negotiations with Jimbo Fisher before deciding to keep Les Miles as the head football coach:
The potential deal to fire Les Miles as LSU’s football coach began to fall through early last week as the school’s president started to intervene even as negotiations between LSU and Florida State coach Jimbo Fisher were ongoing, multiple sources have told The Advocate.
This being Louisiana, politics were in play. Outgoing governor Bobby Jindal lobbied on Miles's behalf, for instance. School president F. King Alexander also worried that more than $30 million it would take to replace Miles and his staff with Fisher and a staff of his choosing—a number that "shocked" the president—would play poorly the next time he needs to go to the state legislature to fight for funding.
And before you jump in about how athletic departments aren't funded with public money:
Private funds from several major donors — not university funds — would have covered much, if not all, of the expense, but Alexander told the [Baton Rouge] Business Report many in the Legislature and in the general public fail to understand that distinction. Also, if Miles were to find another job, the buyout money would have been offset, according to a copy of the contract.
"The public at large really doesn’t differentiate where the money comes from," Alexander told the outlet. "No matter how you explain it, it’s still a $15 million to $25 million decision that needs to be factored into the overall equation."
This matters, because it doesn't take much Google searching to find stories from recent years about LSU planning for "academic bankruptcy", possibly losing hundreds of faculty members, looking at having to turn students away from a popular major to avoid it losing accreditation over having too high a student-to-faculty ratio, and university hospitals falling under duress all because of a lack of funding from the state legislature. It doesn't matter much if boosters are footing the entire buyout cost if the headlines say "LSU spends $30 million to replace its football coaching staff", which they would. Ignorant—or willfully ignorant, or disingenuous—politicians would hold that against the school the next time it has to plead its case for funding.
So Miles came closer to getting the boot than it seemed, if people associated to LSU and people associated with Fisher had already began formal talks. Forces bigger than football may have helped save Miles's job this time, but it's no less an impetus for him to get a better staff in order to make sure that he doesn't need outside help next year.
Our friends at And the Valley Shook have a more expansive exploration of this topic here.