The Sarina Wiegman 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 Women's World Cup final, in the process becoming the only manager to do so with two different nations in the showpiece after steering her home country to the same stage four years ago.
Vlatko Andonovski has stepped down as coach of the United States after the four-time champions produced their worst-ever World Cup performance in Australia, exiting in the round of 16.
Jill Ellis, who was in charge of the US winning campaigns in 2015 and 2019, told reporters in Sydney on Thursday that while a search for a replacement should include diverse candidates, the new coach's gender should not be the decisive factor.
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 Sarina Wiegman takes a well-deserved post-tournament holiday before striking up any conversations about extending her stay.
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 their 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 Wiegman 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."