Mayfield needs big wins to go with the big stats to truly have a shot at reaching their level. After all, White and Bradford got the Sooners to national championship games. Mayfield's potential Heisman run and Oklahoma's push for a slot in the College Football Playoff kick into high gear Saturday night, when the 12th-ranked Sooners (8-1, 5-1 Big 12, No. 12 CFP) play at No. 4 Baylor (8-0, 5-0, No. 6 CFP).
Oklahoma coach Bob Stoops wouldn't move Mayfield into the Heisman conversation just yet.
''That's not for me to say,'' he said. ''A lot here in the next few weeks will matter.''
''The whole season, we knew our last three games were our tougher ones just because of the opponent we were facing,'' he said. ''We've been kind of waiting for this November stretch for a while. It's finally here. It really is one game at a time if we want to go where we think we're capable of going.''
The Sooners have scored 50 or more points in four straight games for the first time since 2008, when Bradford won the Heisman. Overall, Mayfield is slightly better statistically across the board than White was nine games into his 2003 Heisman run. He has fewer yards and touchdowns than Bradford did in his Heisman year, but he has a higher completion percentage, fewer interceptions and more rushing yardage and touchdowns.
White's team in 2003 was unbeaten through nine games. Bradford's team five years later had lost to Texas, which is also the case this season.
Mayfield already has had a few Heisman-type moments. In his second start, he led the Sooners back from a 17-0 deficit against Tennessee to win in overtime. He threw two touchdown passes in the fourth quarter, then ran and passed for a score in overtime to help Oklahoma beat Tennessee 31-24 on the road. The next week, Mayfield set a school record with 572 yards of total offense in a 52-38 win over Tulsa.
Mayfield was reckless at times when he was at Texas Tech, but this year, he ranks second nationally in quarterback rating. In league play, he has completed 72 percent of his passes for 292 yards per game, with 18 touchdowns and just two interceptions.
''What I'm proud of is he's done such a good job of being smart with the football, to go with making great decisions of when to hand off, when to throw it and throwing a great deep ball and all of that,'' Stoops said.
Mayfield's ability to extend plays makes him dangerous. He has rushed for 225 yards and five touchdowns this season.
''We talked about fearlessness, and he's certainly got that,'' Baylor coach Art Briles said. ''He's just a guy that plays with a lot of confidence. He's very accurate, can use his feet, tough kid, and a good football player.''
Mayfield said his confidence comes from trusting the system and his teammates. He knows he needs to relax to be at his best.
''When I'm a little bit too hyped up, I don't settle in and make the routine plays that I should,'' he said, ''It's all about making the routine plays, because I don't have to do anything special considering the guys I have around me. My supporting cast is very talented. I need to go out there and do the routine plays, which is just execute and get the ball out of my hands.''
The only blip for Mayfield was a rough outing against Texas. He completed 20 of 28 passes, but was sacked six times in the 24-17 loss. Baylor nickelback Travon Blanchard said the Bears will try to duplicate some of Texas' success.
''Texas did a really good job with keeping Mayfield uncomfortable in the pocket,'' Blanchard said. ''He never really had that much time to sit and go through his progressions. So if that's something that we can do, that would be tremendous in our game plan.''
Follow Cliff Brunt on Twitter: www.twitter.com/CliffBruntAP