There are AT LEAST three different stories regarding Coach Bryant's successor and probably more that we don't know about.
1) Gene Stallings was supposedly the chosen one and this was suggested by Bryant Jr at the press conference hiring Stallings, when he said, "This is what papa wanted." How much of this was Jr validating the hiring and how much was real - who knows?
2) Ray Perkins was supposedly the chosen one, Bryant's "favorite end." When the insiders told Bryant Perkins was not doing very well in NY with the Giants as head coach, Bryant allegedly blamed the organization of the Giants (Perkins had just taken them to their first post-season since 1963).
On this note, we also have this post from Earle in 2008:
I'll tell you what I heard at the time, and it came from someone personally close to him. You may remember that Coach's public position was that he was staying out of the selection process. If were asked privately, and he was, his second choice was CGS. but his first choice was, in fact, Perkins. The logic was that Coach recognized that the wishbone was passe' - that it had been well dxed and could be defended. One of the reasons he gave publicly for his retirement was that he never wanted to coach kids to pass-block again. Perkins was a respected offensively-minded coach, whereas CGS was the opposite - a defensively-minded coach. He felt that the defense would be fine, but the offense needed revamping totally. I've always accepted this as the truth, both because of who told me and also because it made sense. Whatever the order of choice, it would be unthinkable that Perkins would have been hired without Coach's strong stamp of approval, no matter what he said publicly...
3) "The Uncivil War," a history of the Iron Bowl 1981-94 by one Tide grad and one Auburn grad opens with the tale that Bryant called Pat Dye, who was then coaching Wyoming, to tell him to NOT take the Auburn job. Auburn was trying to land Vince Dooley, who was prepping his team for the Sugar Bowl national title game against Notre Dame. Dooley, who won a ring with Auburn in 1957, turned them down. Dye himself has told this story as well (see below). Allegedly, Dye didn't want to follow Bryant and simply maintain the program, but he also was realistic enough to imply that even if he had wanted it that Bryant had so many assistants out there with better credentials that Dye never could have gotten it.
http://www.al.com/sports/index.ssf/2015/12/pat_dye_reminisces_about_bear.html
Of course, Bryant may also not have wanted to compete against Dye (at Bryant's rapidly advancing age and poor health) for recruits, in part because (for those who weren't alive) Bryant was going to be required by state law to retire after the 1983 season anyway due to his age. Dye tore out on the road in 1981 with a list of recruits (and Bobby Lowder's bankroll no doubt) and basically sounded the trumpet of, "Bryant ain't gonna be there anyway."
And when we blew the recruitment of Bo Jackson........we enabled what happened ourselves.