The Ruben Kazan defender will be put through his paces and checks in Moscow ahead of a deal worth €5m according to reports in Sweden.
Starfelt emerged as a transfer target for Postecoglou at the end of last week with Celtic short of numbers at the back.
Kristoffer Ajer has been linked with a move out of Celtic Park, and Christopher Jullien is still recovering from his knee injury sustained last year.
Stephen Welsh and Leo Hjelde have shown up well in pre-season but combined with Shane Duffy’s return to Brighton following his disappointing loan spell, the Hoops require defensive strengthening and Swedish international Starfelt fits the bill for the new boss.
The 26-year-old ball-playing centre-half has spent two years in Russia, and helped Kazan to fourth in the league last year with the country’s second best defensive record.
His transfer could also net IFK Gothenburg a half-million sell-on fee, say sources in Sweden.
Starfelt is the latest deal to emerge in a busy few days for Celtic who are also poised to announce the signing of Irish youth international Bosun Lawal from Watfrod.
Sources in England say a three-year deal has been agreed for the 18-year-old, compared to Everton midfielder Abdoulaye Doucoure by Vicarage Road legend Troy Deeney, with the cross-border fee totalling around £120,000.
Celtic are also said to be close to a deal with diminutive Israeli winger Liel Abada. The teenager is said to cost around £3.5m from Maccabi Petah Tikva after scoring 13 goals last season.