According to Variety, the Academy received three claims on Wednesday, sparking a probe into his behavior.
The Oscars organisers are investigating its president over reports of sexual harassment.
The Academy received three claims of sexual harassment against Bailey on Wednesday and began an investigation, according to United States celebrity news sites Variety and The Hollywood Reporter. If forced to step aside he would be temporarily replaced by the vice-president, Lois Burwell, a veteran makeup artist.
In a brief statement issued hours later, which did not name the president, the academy said it treated all complaints confidentially to protect all parties.
"The membership committee reviews all complaints brought against academy members according to our standards of conduct process, and after completing reviews, reports to the board of governors".
The academy is not the only awards body to launch an investigation into a senior figure.
Following the slue of allegations made against Harvey Weinstein in October previous year, the Oscars board chose to expel him stating 'sexually predatory behaviour and workplace harassment in our industry is over'.
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The organisation also insisted there would be no further statements "until the full review is completed".
Bailey couldn't be reached for comment. They are the only husband-wife board members in Academy history after founding AMPAS members Douglas Fairbanks and Mary Pickford.
In December, the Academy established a code of conduct which provides that members may be disciplined or expelled for abuse, harassment or discrimination.
Bailey, 75, a married movie director and cinematographer who worked on films like The Big Chill and American Gigolo, was elected president of the Academy in August, 2017.
John Bailey, a veteran cinematographer who heads the film industry's trade group, is facing three harassment allegations, Variety reported on Friday.
Though the position of academy president is unpaid and has traditionally been largely ceremonial, in recent years, it has become more demanding and closely scrutinized.