He took training for the first time on Thursday and has outlined the need to add to the squad before the crucial Champions League qualifier with Danish runners-up FC Midtjylland.
Celtic hope to announce some additions “very, very soon”.
"Look, we're working hard, and it's a challenging time all around the world in terms of signing players and the ability to even watch players," Postecoglou said.
"But the club's working awfully hard, I'm pushing them as hard as I can and we're hoping we can make some announcements very, very soon.
"And certainly, from our perspective, there is some urgency in at least a couple of positions — particularly with this game coming up in four weeks' time."
He added: "It's fairly extensive. It is a season of change. But after a period of success and what happened last year, it was always going to be a period of change.
"But I see it as an opportunity to make changes in the direction I want.
"We're working very hard but with the players we have existing, they will all be treated on merit. For the ones coming in, we'll bring in ones who will fit."
Postecoglou was appointed after a protracted period which saw Celtic set their sights on former Bournemouth boss Eddie Howe.
The 55-year-old admitted to not being bothered about being second or even “fifth choice” as aims to make the most of the “opportunity” to manage the Parkhead club.
He is now tasked with usurping Rangers who cantered to the Premiership title last season, winning it by 25 points.
"I guess what I bring is maybe a different perspective,” he said. "I've had a different journey from many others. Most of my career has been spent on the other side of the world and in internationals.
"Every team I've coached has had a very clear identity.
"I think that was one of the reasons I was chosen. In my 25 years in coaching, my teams have always played a certain way."
He added: "It is a massive challenge. But as a manager you always get opportunities where it's a massive challenge. You don't really walk into places where it's running smoothly.
"It's a process where you have to go through certain steps. If we play our football, we've got a decent chance of success."
Griffiths, Edouard and Ajer
Postecoglou confirmed he has had discussions with Leigh Griffiths, although no deal has been agreed, while he plans to talk to the likes of Odsonne Edouard and Kristoffer Ajer who have been linked with a summer exit.
"They're both just arrived in the country, as have I,” he said. “I'll have discussions with them and we'll make the best decisions for the football club.
"That's where my interest lies. For me, the individuals within that, I totally understand and respect their own wishes but I'll always make decisions in the best interests of this football club because that's my responsibility.
"Whatever I think is going to be best for our football club moving forward, that's the direction we'll go."