Redknapp was widely tipped for the England job when Fabio Capello resigned last year, but the FA opted to approach Roy Hodgson, then in charge at West Brom.
At the time he had expected to be leading England in Euro 2012, Redknapp was sacked by Tottenham and is now manager of QPR in the Sky Bet Championship, having been relegated with the Rs last season.
In his new autobiography, being serialised in the Daily Mail, Redknapp takes a swipe at the FA's senior officials, saying he would not trust their judgement of a manager "if their lives depended on it", arguing: "What clubs have they ever run? Who do they speak to who really knows the game?"
Redknapp added: "This isn't about them giving the England job to me or Roy Hodgson, but English football being run by people who really haven't got a clue. And they get to pick the England manager!"
But Bernstein chaired the four-man party that assessed the candidates to succeed Capello, and he is certain that, in Hodgson, they identified the ideal man.
He said on BBC Radio Five Live: "There were four people who made this decision: one of them was myself, and I've been involved with running Manchester City for 10 years, and chairman of that club for five years, and have since been involved with the FA and Wembley Stadium; Sir Trevor Brooking, who has a lifetime in football and also managed, and done a great deal in football; Adrian Bevington, who has huge football knowledge, amazing football knowledge - he may not have managed but he has; and Alex Horne, the general secretary.
"We spoke to up to 20 people within the game, from right across the game: other managers, players, all sorts of people with great, great knowledge. I believe we did a very thorough job and I was very proud of the way we did it. It was done discreetly and professionally and I believe we ended up with the right choice."
Asked if Redknapp was ever a contender, Bernstein said: "I won't answer that question."
Redknapp now pins the blame for QPR's relegation on the attitude of some of the players at Loftus Road, and claims even Jose Mourinho would not have kept them in the Premier League.
Redknapp took over at Loftus Road in November but was unable to prevent a slide into the second tier, with big-money players failing to produce the required performances and results.
Even a January spending spree was unable to halt the slide.
Redknapp says in his book: "Jose Mourinho would not have kept QPR up last season. The squad I inherited in November last year was poorly balanced, undisciplined and short of confidence.
"The directors and owners were nice people but they were naive in football terms and I think certain people they had trusted - agents and advisers - had let them down quite badly."
Assessing the nature of players on his staff, Redknapp said: "The attitude stank. Attitude towards the game, attitude towards training. I can't remember a worse one - and behaviour like that cannot be altered overnight."
He added: "Big wages made the players very arrogant and contemptuous. They would rather come in late every day and just pay the fine than behave in a professional manner. Getting them in was daily aggravation."