Gallop said he had discussed the matter with Postecoglou after rival media groups News Ltd and Fairfax said the coach would quit regardless.
Football Federation Australia chief executive David Gallop also hinted at possible changes when he said the Socceroos needed to "lock in our set-up" after next month's play-off with Honduras.
Gallop said he had discussed the matter with Postecoglou after rival media groups News Ltd and Fairfax said the coach would quit regardless of whether Australia qualify for Russia 2018.
The Asian champions, who have reached the last three World Cups, needed extra time in Sydney on Tuesday to beat Syria 3-2 on aggregate and reach the final, two-legged play-off with Honduras.
"I am aware of reports today suggesting that I will step down as head coach of the Socceroos next month," Postecoglou said in an FFA statement.
"My sole focus is on preparing the team for our final two qualifying matches. I will not let anything compromise the team's journey on getting to a fourth consecutive FIFA World Cup."
Gallop also said he wanted to keep the focus on the play-off games in November, whose dates are yet to be confirmed by world body FIFA, rather than Postecoglou's future.
"We have every confidence that Ange and the players can get the job done," Gallop said, referring to the play-off.
"Beyond that, should we qualify, there is a period of some months until the World Cup and we agreed that we will need to lock in our set-up as soon as possible to maximise our preparation time," he added.
News Ltd's Herald Sun said Postecoglou was irritated by criticism during Australia's laboured qualifying campaign, while the Sydney Morning Herald said he wanted to join a club overseas.
The former Brisbane Roar coach took over the Socceroos after Holger Osieck's sacking in 2013, and signed a five-year contract that is due to expire after next year's World Cup.
"He hasn't got a job lined up, he would be flying blind," the Sydney Morning Herald quoted a source as saying.
"It's simply the fact he has been in this job for more than four years, he has been to a World Cup, he has helped raise the standard of the Australian game. He has brought the country its first major silverware with the Asian Cup win.
"He now wants to return to the day-to-day involvement at club level overseas. He feels he has raised the reputation of Australian players on the pitch and now he wants to do the same for Australian coaches."
Under Postecoglou, Australia exited the 2014 World Cup at the group stage but then lifted the Asian Cup the following year, their first major trophy.
The former Brisbane Roar and Young Socceroos coach gave few signs that his departure may be imminent after Wednesday's tense victory over Syria in Sydney.
"I've had confidence all the way. The nature of this qualifying process is we’ve done things in a difficult way the whole time," Postecoglou told reporters.
"Maybe it's not a bad thing. I've got full faith in these guys.
"Whenever it's needed, they keep delivering and keep believing and working hard. I've got no doubt come those two games in November, they'll do what's needed again."