The complaint stems from allegations that Zille's son, Paul Maree, had unduly benefitted from his relation to the premier.
Paul Maree is one of the directors of Paper Video, a company that loaded teaching material onto tablets purchased via the Education Department. The devices were returned to the department.
"As the ANC we believe the premier’s conduct amounts to a breach of the code of ethics for members of the provincial cabinet, we say so because it is clear that the preferential treatment given to Paper Video, of which her son is one of two directors, would never have been given to any other aspirant entrepreneur," Dugmore said.
"Information which we have received and have provided to the Public Protector indicates the direct and improper involvement of the premier in email exchanges which led to 150 government tablets being procured for the use of Paper Video."
In a published opinion piece, Zille wrote that Paper Video was allowed usage of the 150 tablets, which was part of a consignment of 480 devices ordered for the Year Beyond (YeBo) Programme.
The devices were purchased through the Department of Education, because they were going to become the property of schools where the YeBo interns were working, Zille wrote.
But Dugmore believes Zille acted improperly.
"The Premier has not played by the rules, and this is a clear case of nepotism and we trust the Public Protector’s investigation will involve the calling of witnesses who we believed will provide compelling evidence that the premiers have beach the code of ethics."
Public Protector spokesperson Cleopatra Mosana said: "I can confirm that we have received a complaint from MPL Cameron Dugmore, and the complaint is being assessed."
Zille wrote in the article that Maree prepared a matric programme, and volunteered to give free workshops to disadvantaged students ahead of the matric exams in 2014.
"An expert in the department had evaluated the quality of the material, the purchase of 150 out of the 480 tablets was expedited to make sure they arrived in time for the workshops. I supported speeding up the process, just as I support the expediting of every procurement process as long as no rules are broken."
Zille said Maree didn't make any money out of the project, raising the necessary money from donors, and convening workshops with the permission of the relevant schools.
Zille's spokesperson Michael Mpofu said Dugmore incorrectly claimed this was an "unsolicited bid".
"Legal opinion provided to us is that this was not an unsolicited bid. It is comparable to a private doctor who offers their time and expertise to work in a state hospital using the equipment available.
"The only people who benefitted from the workshop was the kids themselves, who got free DVDs and books with hundreds of hours of teaching on them.
"The Public Protector is therefore welcome to investigate the matter," Mpofu said.