The Scottish champions host the Estonians in the first leg on Wednesday while Cluj host Maccabi following Monday’s draw.
If Neil Lennon’s side lose, they drop into the Europea League qualifiers.
In that scenario, they will play the winners of Shkendija of Macedonia and Luxembourg’s F91 Dudelange.
It was Shkendija who Nomme Kalju defeated on away goals in the first round of Champions League qualifying to set up their tie against Celtic.
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Lennon’s side will be favourites to progress against the Estonians and set up a possible tie against a side led by a head coach familiar to British football fans.
Former Chelsea and Romania right-back Dan Petrescu has returned for a second spell as Cluj head coach after leaving Guizhou Hengfeng following their relegation from the Chinese Super League in his first season in charge.
Should they beat Nomme Kalju, Celtic will travel to either Romania or Israel for the first leg on 6/7 August, with the return on 13 August.
Meanwhile, should Welsh champions The New Saints defeat Danish champions Copenhagen, they will face either Crvena Zvezda of Serbia or HJK Helsinki of Finland.
Republic of Ireland’s Dundalk will take on Sutjeska of Macedonia or APOEL of Cyprus should they prevail against Qarabag of Azerbaijan.
The 51-year-old Petrescu inherits a side who won Liga 1 for the fifth time last season – and second season in a row.
Venezuela striker Mario Rondon, who has switched from Liga 1 rivals Gaz Metan, and defender Mike Cestor, who started his career with Leyton Orient before joining Woking and was with Astra Giurgiu, are among Petrescu’s summer signings.
Also among his playing staff are former Aston Villa midfielder Yacouba Sylla, one-time Charlton Athletic striker Marius Tucudean and Lithuania goalkeeper Giedrius Arlauskis, who had a spell with Watford.
Maccabi topped the Israeli Premier League for a 22nd time last season, but it was their first title since 2015.
In their first season under former Serbia, Partizan and AEK Athens midfielder Vladimir Ivic, they swept to the title, 22 points ahead of Maccabi Haifa, with defending champions Hapoel Be’er Sheva down in fourth.
Among their summer signings is winger Nick Blackman, who has arrived on a free transfer from Derby County after starting last season on loan with Maccabi before a spell with Sporting Gijon, and 19-year-old Panama striker Eduardo Guerrero from Universitario.
Also in the squad are Israel striker Itay Schechter, who had a spell on loan with Swansea City, while right-back Andre Geraldes has arrived on loan from Sporting Lisbon.
The Red and Blacks, who lost on away goals to Aberdeen in the first round of Europa League qualifying in 2015, won the Macedonian First League for the second year running.
Dudelange, who won the Luxembourg National Division for the fourth season running, have dropped into Europa League qualifying after losing to Valletta of Malta on away goals in their Champions League opener.
Champions League third qualifying round draw
Cluj (Romania) or Maccabi Tel-Aviv (Israel) v Celtic or Nomme Kalju (Estonia)
Sutjeska (Macedonia) or APOEL (Cyprus) v Dundalk (Republic of Ireland) or Qarabag (Azerbaijan)
PAOK (Greece) v Ajax (Netherlands)
Saburtalo (Georgia) or GNK Dinamo (Croatia) v Ferencvaros (Hungary) or Valletta (Malta)
Crvena Zvezda (Serbia) or HJK Helsinki (Finland) v The New Saints (Wales) or Copenhagen (Denmark)
Maribor (Slovenia) or AIK (Sweden) v BATE Borisov (Belarus) or Rosenborg (Norway)
Istanbul Basaksehir (Turkey) v Viktoria Plzen (Czech Republic) or Olympiakos (Greece)
Krasnodar (Russia) v Porto (Portugal)
Club Brugge (Belgium) v Dynamo Kyiv (Ukraine)
PSV Eindhoven (Netherlands) or Basel (Switzerland) v LASK (Austria)