Australia's 6-0 loss to France on Friday resulted in the dismissal of German Holger Osieck, who oversaw a disappointing streak of losses to Japan, China and Brazil in recent months.
While refusing to rule out Guus Hiddink - who led the Socceroos at the 2006 finals - or former Liverpool boss Gerard Houllier, Gallop said nationality did not matter so much as the new coach's ability to restore success to the team.
"It certainly makes sense to have an Australian coach, but whether that makes sense right now is something we need to look at seriously over the next few weeks," he told reporters.
"We're always exploring options and at the moment that's all we're doing. It's not appropriate to go into who has been contacted and who hasn't but obviously some feelers have been put out.
"This is about the need to rejuvenate, regenerate our team. We've got a lot to look forward to in the next months and years.
"Obviously Brazil is important as is the 2015 Asian Cup and ultimately we want to be at the next World Cup as well. This is a decision taken in the long-term interests of the team.
"We'll hopefully have some definitive answers on that over the next few weeks."
Hiddink, who left Russian club Anzhi Makhachkala in July, has enjoyed notable stints leading Asian outfits at the World Cup, guiding South Korea to an unlikely fourth-place finish in 2002 before his Australia side only fell 1-0 to eventual winners Italy in 2006.
Houllier, meanwhile, took charge of France at senior and youth levels between 1992 to 1997, before moving into club management with sides including Liverpool, Lyon and Aston Villa. He is presently unemployed.