Several years on, Bledsoe has the chance to take his impersonation a stage further. On Sunday he will attempt to win the Super Bowl in the Louisiana Superdome, New Orleans - just as McMahon did 11 years ago. Ironically, McMahon will be on the opposing sideline as a member of the Green Bay Packers.
The maverick McMahon and the drill sergeant figure of his Bears head coach, Mike Ditka, formed the NFL's odd couple of their day, a mantle that has since passed to Bledsoe and his Patriots mentor Bill Parcells.
Bledsoe is the kid with the golden arm, a precocious talent that comes with a laid-back West Coast attitude. Four years into his NFL career, fame and fortune - he is two years into a six-year, $42m (pounds 25.5m) contract - have already been earned. In fact, Bledsoe appears to represent everything that Parcells, East Coast tough and proud of it, hates in football. The New York Giant teams that Parcells coached to two Super Bowl victories did it his way; tough defense, a solid running game and sound fundamentals. Fat passing statistics and even fatter contracts mean nothing to Parcells; championship rings are the only currency that counts.
At the Giants, the quarterback passed when Parcells decided it was the sensible option, not because it was the only option. It must gnaw at Parcells' football beliefs every time the Patriots win on the back of a 24-year- old going out and throwing 40-odd passes. But that is how it has been over the past four years. As Bledsoe goes, so go the Patriots.
Despite the addition in the past two years of a running threat in the shape of Curtis Martin, there have been few occasions when Bledsoe has been able to get away with playing as indifferently as he did in the Patriots' AFC Championship victory over the Jacksonville Jaguars. It is unlikely that the Patriots' chances of beating the Green Bay Packers in New Orleans will survive another such performance, especially if the Packers shut down the New England running game as effectively as they stymied the Carolina Panthers' ground attack in the NFC Championship game.
Even Parcells knows he cannot win without Bledsoe, despite the fact that he told his play-caller at the beginning of the 1994 season, their second season together: "Just remember one thing: I don't want a celebrity quarterback on my team. I hate celebrity quarterbacks."
Which suits Bledsoe just fine. Unlike the publicity-conscious McMahon, Bledsoe never went looking to be the focus of attention. The spotlight naturally picked him out because of his ability to hurl a pigskin vast distances in the right direction. He even chose Washington State University because it offered him the chance to play top-level college football away from the media circus that surrounded some of the more illustrious schools who had shown an interest.
When he arrived at the Patriots in the summer of 1993 as the much-heralded first pick of the draft, Bledsoe found himself having to fetch drinks for his head coach and was generally treated like the most insignificant player on the team. Bledsoe's father, Mac, says: "It was great for Drew. By Parcells being all over him, it allowed Drew to be one of the guys, to fit into the chemistry of the team and be accepted by guys who might have resented him."
Bledsoe's relationship with Parcells is interesting in that it is so different to that with his own father, a teacher and high school assistant football coach. Mac Bledsoe, who counsels parents on self-esteem for children, says: "I believe that in coaching and in teaching, kids have to find their own way."
The young Bledsoe was encouraged to do his own thing, grow in his own way - a philosophy which does not quite fit into the Parcells method of man management. Like Ditka, with whom he shares a history of heart trouble, Parcells is a renowned ranter and raver on the sideline. Bledsoe says: "Sometimes I just want to scream at him: 'Shut up!' We're different. I don't believe he's going to make me play any better by yelling at me the way he does, but that's his way, and he's won that way. The reason we can co-exist is that we both have the same goal in mind. And I think we both respect each other.
"Parcells is a smart guy but he turns the pressure up in practice when we're not winning. That's the hard part. All the coaches are so uptight. You almost never see me fired up and ticked off. I still look at this as fun. So the nice thing is that, when I'm on the field, I'm in that sanctuary. I'm in that place where nobody can touch me. Out there everything's under my control."
Parcells might not be comfortable with how often Bledsoe has to throw the ball but, as a coach who loves good technique, he can have no complaints about the way in which he does it. Bledsoe is a pure pocket passer. Not for him, the scrambling around in the backfield. When he is on form, the 6ft 5in Bledsoe steps up, stands firm and tall and launches his bombs. "He never threw the ball incorrectly," Mac says. "He's a clinic in how to throw. You have to see it at field level to appreciate it. It's not his arm, it's his body the arm is just along for the ride. Watch him throw an out pattern toward you. I mean, it's coming. It whistles."
If anyone in the NFL is ever going to get anywhere near the passing records set by the Miami Dolphins' Dan Marino, Bledsoe may be the one. He has already overtaken Marino as the youngest player to pass for 10,000 yards, achieving the feat before his 24th birthday. In 1994, he set a single-season record by attempting 691 passes. A two-time Pro Bowl selection, Bledsoe this year surpassed 4,000 yards for the second time in the past three seasons.
But records mean nothing without a Super Bowl ring. Marino had his shot in the second year of his career and, in the uncertain world of the NFL, has never been back. Drew Bledsoe, therefore, should need no urging to make the most of the opportunity that presents itself in New Orleans on Sunday. But in case he does, Bill Parcells will be yelling from the sidelines. Just to make sure the message gets through.