The 56-year-old has already bolstered his squad with four new players this summer, including winger Jota and centre-back Cameron Carter-Vickers, who have both joined the club on permanent deals after enjoying successful loan spells at Parkhead last season.
The Hoops have also signed goalkeeper Benjamin Siegrist on a free transfer from Dundee United and Argentinian left-back Alexandro Bernabei from Lanus for around £3.9m.
However, Postecoglou is still looking to improve the depth of his squad ahead of the 2022-23 campaign and is hopeful that a couple more players will join the club next week.
Speaking to reporters, via the Daily Record, after their 2-2 pre-season friendly draw at home to Blackburn Rovers on Saturday, the Australian said: "We're definitely going to add a couple of players.
"We're working on that at the moment but would expect to do that in the next seven days.
"We have had to be patient just to get the ones that we want and that was much more important than rushing people in.
"I'm pretty confident we're close to the end point in bringing in a couple which will help.
"The squad is looking really good and strong but on days like today we had two or three missing and we want to make sure we don't lose our effectiveness when a couple are missing."
Postecoglou believes the addition of Champions League football to their schedule this season is likely to place an extra burden on his first-team squad, and he is now keen to add quality to his side in order to compete at the highest level across all competitions on a regular basis.
"The reality is that we'll play close to 60 games this season," the Hoops boss added. "The Champions League on a weekly basis will add stress to the squad which is why I'm keen to get a couple more in, to make sure we're strong and robust.
"Last year it affected our ability to perform at a really high level. I'm not saying we'll have a bigger squad but it'll be more robust with quality."
Celtic have reportedly identified Paris Saint-Germain midfielder Edouard Michut as a potential target this summer.
A report from the Independent, however, claims that the Hoops are unwilling to meet the French club's €6m (£5.1m) asking price of the 19-year-old.
The Scottish Premiership champions have two more pre-season friendlies against Legia Warsaw and Norwich City before beginning the defence of their top-flight title at home to Aberdeen on July 31.