Southgate has taken his side to the 2018 World Cup semi-finals, the Euro 2020 final and the 2022 World Cup quarter-finals in Qatar.
However he told German newspaper Bild that Euro 2024 could be his “last chance” to win a trophy with England.
Bullingham denied speaking to any potential successors, but said: “I’ve seen some things about have we got a plan, or haven’t we.
“Any organisation really has a succession plan in place for their top employees, and we are no different to that.
“That’s really all I want to say on that topic. Just because I want to respect Gareth and the team. They are very focused on the tournament, and we want to be supporting them with that focus. Clearly, you plan lots of different scenarios for all your top employees all the time.”
When asked if Southgate’s future would depend on how England did in Germany, Bullingham said: “I know why people would love for there to always be a really arbitrary level. But I don’t think you can set one for any tournament with any manager that you judge, because you could go further but be playing poorly, or have a really unlucky result where you get a couple of red cards.
“So I think setting an arbitrary figure isn’t the right way to go. I think we step back and look at everything after the tournament, see how he feels, see how we feel and see how the tournament has gone.”
When asked if he would want any successor to be a homegrown manager, Bullingham said: “We have two senior coaches. One of them is English, one of them [England women’s boss Sarina Wiegman] is not. Any federation in the world would always want to have a pool of top homegrown talent playing and managing at any time.”