While some new arrivals are expected before the first leg of the Champions League second qualifying round tie against Midtjylland at Celtic Park on July 20, Postecoglou must largely rely on a core of incumbent personnel for his initial assignments.
Some are tried and trusted, including likely new captain Callum McGregor and winger James Forrest whose vast experience at European level will be invaluable for Postecoglou as he tries to hit the ground running in the job.
The one-year contract extension agreed with Leigh Griffiths also provides a proven striking talent, albeit one whose fitness levels and conditioning have been questioned.
But the 55-year-old Australian coach has decisions to make over a raft of other players in the first team squad who are still under contract but whose futures at the club remain uncertain.
Can Postecoglou get the best out of those who have struggled to make an impact so far? Could there be a second chance for those whose time at Celtic appeared to be up?
The Swiss international striker arrived to considerable fanfare and at a significant cost when Celtic signed him from West Ham United on a four-year deal for £5 million last summer.
But despite scoring a winner on his debut against Dundee United, Ajeti was unable to sustain that momentum during a season when he too often appeared to be lacking in physical durability.
The 24-year-old was among those under the watchful eye of Postecoglou in pre-season training this week and will have the opportunity to stake his claim for more regular involvement when Celtic kick off their schedule of friendly fixtures against Sheffield Wednesday next Wednesday.
Like Ajeti, Greek international goalkeeper Barkas was a high profile recruit 12 months ago but has toiled to vindicate the £4.5 million fee Celtic paid to AEK Athens.
The 27-year-old was believed to have the pedigree and ability to fill the void left by the failure to complete a permanent deal for Fraser Forster and the questionable decision to allow Craig Gordon to depart.
But Barkas could not meet those standards last season and by October had lost his place in the first team to Scott Bain. Perhaps the Greek-born Postecoglou can inspire Barkas to find the form required to become Celtic’s number one in the new campaign.
The French midfielder’s obvious talent hasn’t always been matched by the desired levels of commitment during his four years as a Celtic player.
After an unsuccessful loan spell with Marseille in the second half of last season, it looked as if Ntcham had found the exit door in the shape of a move to AEK Athens this summer.
But with a year still remaining on his contract, negotiations over a fee have cast doubt on that move. It remains to be seen if Postecoglou will seek to keep the 25-year-old on board for the new journey Celtic are undertaking.
The Ivory Coast international striker spent last season on loan at Toulouse, scoring ten goals in 31 appearances for the French club who narrowly missed out on promotion to the top flight when they lost on away goals to Nantes in the play-off final.
Toulouse are keen to extend Bayo’s stay but are unable to meet any transfer fee demand from Celtic for a player who still has a year remaining on his contract.
Bayo has managed just two goals in 19 games for Celtic since his £1.8 million move from Slovakian club Dunajska Streda in January 2019.
The Ukrainian international winger still has two years remaining on his Celtic contract, having signed from Karpaty Lviv in his homeland in January 2019 for around £1.5 million. It was a deal which was greeted with a degree of indifference at the time by then manager Brendan Rodgers.
Shved has made just three substitute appearances for Celtic, scoring on his debut in a Champions League qualifier against Nomme Kalju in July 2019.
Last season, the 24-year-old was loaned out to Mechelen, scoring four goals in 22 appearances for the Belgian side who are hopeful of securing his services again for the new campaign.
Shved had been tipped by Ukraine head coach Andriy Shevchenko to become a major star for his country but now needs clarity on his club future before he can revive his international career.
Could there possibly be any way back at Celtic for the Belgian left-back whose mindless breach of coronavirus protocols last August was one of the first of many calamities in a miserable season for the deposed champions?
The 26-year-old, signed from Rapid Vienna for £3 million in the summer of 2019, is certainly regarded as persona non grata by many of the club’s supporters.
Bolingoli was loaned out to Turkish club Basaksehir for the remainder of last season but managed just 12 appearances for the Isanbul outfit who have confirmed they will not be pursuing a permanent deal for the player.
Another Turkish side, newly promoted Altay Izmir, have expressed an interest in Bolingoli who still has two years remaining on his contract at Celtic.
Unless Postecoglou decides he wants to give Bolingoli a clean slate in Glasgow, it’s unlikely Celtic would be too difficult to deal with.