Saban was not at home on that day 25 years ago, but his wife answered the phone, spoke to Meyer and recommended to her husband that he consider him for his staff.
"I was so kind of caught up and busy in what I was doing I never really followed up on that," Saban said. "And, obviously, that was a huge mistake on my part, because he's a fantastic coach."
Meyer climbed the coaching ladder to Bowling Green and Utah before guiding Florida to two national championships, and then revitalizing Ohio State.
"There's a lot of mutual respect there," Meyer said.
On Thursday night at the Superdome, Saban and Meyer will coach against each other for the fourth time when Alabama plays Ohio State in the Sugar Bowl, a College Football Playoff semifinal.
The winner plays Oregon or Florida State for the championship on Jan. 12 at AT&T Stadium in Arlington, Texas.
Alabama is 12-1, its only loss coming against Mississippi on Oct. 4. Saban said his team had "sort of been in a playoff of our own kind," since the defeat.
"Not only did everyone sort of discount this team totally and completely," he said, "it wasn't just that this team wasn't going to make it, it was that the whole era of what we've been able to do at Alabama was done, gone, didn't work anymore and all that.
"So our players really responded to the loss."
Ohio State rebounded from a Sept. 6 defeat by Virginia Tech — the Buckeyes' first loss in a home opener since 1978 — to win 11 consecutive games and finish 12-1.
And the Buckeyes did it by mining the quarterback depth chart.
Braxton Miller suffered a season-ending shoulder injury during training camp. J.T. Barrett played outstanding in his place, but suffered a leg injury in the regular-season finale against Michigan.
Cardale Jones finished that game and started the Big Ten Conference championship game against Wisconsin, leading the Buckeyes to a 59-0 victory.
Asked Wednesday how comfortable he was with Jones and the quarterback situation, Meyer told reporters, "If you asked me a year ago or six months ago, I would have looked at you … like you've got six heads. But it's a much different story right now. He's had a great week of practice."
Jones, a 6-foot-5, 250-pound third-year sophomore, said he would not be overwhelmed by the moment.
"I don't think that they will do anything that we're not prepared for," he said of Alabama.
Crimson Tide safety Landon Collins indicated Alabama would try to take advantage of Jones' inexperience.
"We definitely feel we can rattle him," Collins said.
Saban is 2-1 against Meyer.
Florida defeated Alabama in the 2008 Southeastern Conference championship game. But Alabama won in the 2009 SEC championship game, and a 2010 regular-season game at Gainesville, Fla., in 2010.
In previous years, a bowl appearance would have marked the final game of the season.
But the winner Thursday will play one more.
"It is a much different feel than, 'this is it,' because it's not it," Meyer said. "Everybody knows there's something left."