Thomas Tuchel will become England's third overseas head coach for the men's team when he starts work on January 1, raising questions over the strength and depth of top-level English coaches.
FA boss Bullingham said there were English candidates among the "approximately 10" people interviewed for the job, but former England defender Gary Neville claimed the decision to go for a foreign coach was a "big blow" to domestic managers.
The job profile published by the FA on July 19 - three days after Gareth Southgate's exit - said the governing body was looking for someone with "a strong track record delivering results in the Premier League and/or leading international competitions". No English boss has won the Premier League, and the last Englishman to win the European Cup was Joe Fagan back in 1984.
Bullingham sought to give his backing to the work being done to develop homegrown talent at the national football centre at St George's Park, which he said had been "a really good success".
"Our pathway is really strong, both from a coaches and players' point of view," Bullingham said.
"There are a lot of fantastic young coaches around and obviously (Tuchel's assistant) Anthony (Barry) is one of those.
"I think any federation in the world that is looking to hire a senior manager, clearly you would love to have five to 10 domestic candidates who are coaching clubs in your domestic league, challenging and winning honours in your domestic league and European football. We are not quite in that place at the moment.
"In the background, we have got to keep helping our young coaches to get the best opportunities they can and to get them good opportunities at clubs. We would love to have more English coaches managing in the Premier League, for example. I think there is a balance there."
Other than Barry, it is still unclear which other English coaches will be part of Tuchel's set-up. One name not mentioned by Tuchel or Bullingham at the German's unveiling on Wednesday was former England, Arsenal and Chelsea full-back Ashley Cole.
Cole joined the England coaching set-up full time last month and had been working with the seniors alongside Lee Carsley, who will revert to the under-21s after England's Nations League matches in November.
Asked about his coaching set-up, Tuchel said: "There will be a few more people in the backroom staff but we will keep the group very small because I always want to acknowledge the potential and the quality in the staff of the FA.
"We want to work together and we want to be open but it is important to have a streamlined process, people at your side who watch the game and the behaviour, the patterns through your lens.
"Anthony is the key figure in that, as my assistant coach. We work now for the third time together and I'm very happy to have him at my side."