The former rugby league international, who played the first five of his 66 tests on the wing before switching to fullback in 2013, returns after missing a loss to New Zealand and last week's 23-18 victory over South Africa with an ankle injury.
"I think Dane's played pretty well at fullback and I'd like to give him an opportunity to have another go," Cheika told reporters in Queensland.
"Israel, first game back, just a slightly different look ... we can set up a few things for him in that position and give it a go.
"I think everybody knows how highly I regard (Folau) at 15 and I'm sure he'll want to get that jersey back ... it's just something different, see how we go..."
David Pocock also returns to the back row at number eight having missed the win over the Springboks with a neck injury, one of three players along with Folau and lock Adam Coleman forced to withdraw after Cheika named his side.
Coleman, who pulled out to be at the birth of his first child, returns on the bench with Izack Rodda keeping his place in the starting second row with Rory Arnold.
The starting side is otherwise the same one that Cheika wanted to put out against South Africa.
Kurtley Beale retains the flyhalf role with Matt Toomua, who scored 15 of Australia's 23 points against the Springboks, his playmaking foil at inside centre and Bernard Foley again warming the bench.
The front row of Allan Alaalatoa, Tatafu Polota-Nau and Scott Sio also remains intact with prop Sekope Kepu looking set to win his 97th cap as a replacement after being dropped last week.
Pete Samu, who stood in for Pocock at the back of the scrum in Brisbane, is also on the bench as back row cover along with Nick Phipps, who comes in for Joe Powell as the back-up to scrumhalf Will Genia.
The Wallabies snapped a four-match losing streak last weekend after opening their Rugby Championship campaign with back-to-back heavy losses to New Zealand.
Team: 15-Dane Haylett-Petty, 14-Israel Folau, 13-Reece Hodge, 12-Matt Toomua, 11-Marika Koroibete, 10-Kurtley Beale, 9-Will Genia, 8-David Pocock, 7-Michael Hooper (captain), 6-Lukhan Tui, 5-Izack Rodda, 4-Rory Arnold, 3-Allan Alaalatoa, 2-Tatafu Polota-Nau, 1-Scott Sio.
Replacements: 16-Folau Faingaa, 17-Sekope Kepu, 18-Taniela Tupou, 19-Adam Coleman, 20-Pete Samu, 21-Nick Phipps, 22-Bernard Foley, 23-Jack Maddocks.
(Reporting by Nick Mulvenney; Editing by Amlan Chakraborty)
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