The Los Angeles Times reports Chief Michel Moore says investigators will try to determine how the image may have come into the workplace and who may have been involved.
The officer who made the complaint will be interviewed Monday.
Floyd, a Black man, was killed by Minneapolis police on May 25, 2020, after an officer knelt on Floyd's neck while he said, "I can't breathe." His death spurred protests across the country.
An email posted on Twitter Saturday by journalist, political commentator and activist Jasmyne Cannick appears to indicate the photo and a possible post on social media originated out of the department's Harbor Patrol Division.
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"I just received word through the chain of command of a complaint that was generated on an inappropriate online post from a Department member," Captain Jay Mastick of the Harbor Patrol Division wrote. "The post depicts a photo of George Floyd, with a caption, 'you take my breath away' in a valentine format.
"Chief Moore was disgusted by the post and directed that a Department complaint be generated. "...Messages with inappropriate content may also result in discipline. I ask that you maintain your professionalism and use judicious restraint while posting online because it will ultimately reflect on you and your position with the Department."
Department officials said Saturday that "any actual postings in the workplace" had not yet been confirmed.
"We have raised the apparent existence of the image and directed commands to survey the worksites for it," the department tweeted.
Meanwhile, LAPD HQ tweeted at 8:30 p.m. Saturday: "The Department has become aware of allegations that an image was being passed around the department and this image was in the workplace. There are also allegations that the post with the image was authored by a department employee.''
The tweet continued: "A personnel complaint has been initiated and we are pursuing each allegation including interviewing the department member who brought it to our attention.
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"At this point the Department has not identified any actual postings in the workplace or identified that it was in fact our department employee who created the image. We have raised the apparent existence of the image and directed commands to survey the worksites for it.
"If found any employee or supervisor is directed to take possession and identify those present. The Department will have zero tolerance for this type of behavior.''
The Associated Press and City News Service contributed to this report.