The 53-year-old’s stock as a serial winner has risen steadily since securing the European championship trophy with her native Netherlands in 2017, then doing the same with England last summer.
She has now guided England to a first-ever World Cup final, in the process becoming the only manager to do so with two different nations in the women’s showpiece after steering her home country to the same stage four years ago.
Asked if Wiegman could be seen as a potential successor to Gareth Southgate, Bullingham said: “I think it’s a bit disrespectful of the Lionesses to project it as a step up. People always say it is ‘the best man for the job’ or ‘the best Englishman’.”
“Why does it have to be a man? I think our answer is always it’s the best person for the job. We think Sarina is doing a great job and hope she continues doing it for a long time.”
Pressed as to whether England was ready to have a woman in the top men’s seat, he added: “I think football is behind other sports in terms of lack of female coaches at the top level, and that has to change.
“Do I think Sarina could do any job in football? Yes, I do. I’m really happy with the job she’s doing and I hope she stays doing that job for a long time. If at some point in the future she decides she wants to move into the men’s game, that would be a really interesting discussion but that’s for her, right?
“I don’t think we should view it as a step up. If she decides at some point in the future to go in a different direction, I think she’s perfectly capable.
“If and when we get a vacancy in either of our senior men’s or women’s manager positions, we would go for the best person for the job, which would be the best person capable of winning matches.”
Wiegman’s current contract runs out in the summer of 2025, which would see her through England’s European title defence, with next summer’s Paris 2024 Olympics a possibility – though not a guarantee – should the new Nations League result in a qualification for Team GB.
The rampant rumour mill has Wiegman shortlisted as a potential candidate to replace United States boss Vlatko Andonovski, who is expected to step down after the double-defending champions were knocked out by Sweden for a worst-ever last-16 finish.
Wiegman has a strong affinity for the United States, where she played for the University of North Carolina Tar Heels and was awed by the infrastructure that already existed around women’s football in late-1980s America.
But asked if the FA would reject an approach should the United States come courting the three-time FIFA Best award winner, Bullingham instantly replied: “100 per cent. It is not about money. We are very, very happy with her and we feel she is happy.
“We’ve seen lots of rumours, and look, she is a special talent. We know that. From our side, she’s obviously contracted through until 2025. We think she’s doing a great job. We’re obviously huge supporters of her and I think hopefully she feels the same way.”
Bullingham said the FA would wait until after Wiegman takes a well-deserved post-tournament holiday before striking up any conversations about extending her stay at St George’s Park.
While Bullingham believes Wiegman could have any job in football, he admitted it could still be some time before an England women’s manager would be compensated equally to his or her men’s counterpart.
He added: “I think over time, I think there’s where you’ve got to get to. If you look at the disparity in the market and the income coming in, that’s why you’ve got a difference.
“I would say that Sarina is, within the market she operates, well-paid. And if you look at the comparison in the men’s game, it’s a different market. I really want those markets to merge, over time, and I think that’s where you’ve got to go, but we’re not there yet.”