The project has been in development since 2009 with Jackman. The contemporary musical is set to be scripted by Jenny Bicks ("Sex and the City") and produced by Chernin Entertainment, Laurence Mark and Jackman.
Fox had no comment on Efron's possible attachment, reported Variety.
Michael Gracey is attached to direct. Jackman will play the showman with a specialty in hoaxing a gullible public as he creates the three-ring Barnum & Bailey Circus that made him famous.
Barnum had gained fame as a 19th Century impressario with tours of Tom Thumb and the singer Jenny Lind; he began calling it the "Greatest Show on Earth" near the end of his career.
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If Efron's deal go through, it would serve to solidify the project and give it a shot of momentum.
Efron is having a busy summer with "Neighbors 2: Sorority Uprising" bowing in May and the Fox comedy "Mike and Dave Need Wedding Dates" opening next month.
Efron is also set to star in Paramount's comedy tentpole "Baywatch" which opens next May and also stars Dwayne Johnson and Priyanka Chopra.