Football Association director of football development Sir Trevor Brooking, who worked closely with chief executive Brian Barwick throughout the recruitment process, will liaise with new manager Fabio Capello over the appointment of an Englishman to his back-room team.
Boothroyd told BBC Radio Five Live Sportsweek Programme: "It is irritating. But the game is global and they have to pick the best coaches they think can do the job for their team.
"If the top four keep winning, it is not going to give anyone else a chance because they cannot break into that and it is very difficult."
Boothroyd believes working alongside Capello would allow coaches to take on board a great deal.
The Watford boss said: "If I was in a position where I was picking the England manager, I would not just pick the man who is going to go in now, I would pick the man who is going to go in after him.
"He should be put alongside and groomed, because that legacy needs to be left.
"It would be all about listening and learning."
Boothroyd added: "I would love to develop myself by watching, learning and getting involved in any way.
"I think any English coach would love to do whatever they can to make sure our national side is successful. We all want to do that."