The request for the opaque bags, which would cover the images of scantily clad women on the front covers of the magazines, is aimed at protecting children from seeing sexualised imagery.
The Co-operative has already introduced opaque screens on shelves to conceal most of the magazines' front covers, in accordance with guidelines drawn up by the Professional Publishers Association, in association with the National Federation of Retail Newsagents and endorsed by the Home Office.
It is now asking publishers, including IPC Media and Bauer Media, to provide their own bags, saying that it is responding to concerns by its customers about the over-exposure of children to the overt sexual images on front covers.
The Sport newspaper – also known for nearly naked pictures – has already agreed to deliver all editions in modesty bags after the deadline, after discussions with the retailer.
Steve Murrells, chief executive of retail for The Co-operative Group, said: "Whilst we have tried to mitigate the likelihood of young children seeing the images with a number of measures in-store, the most effective way of doing this is for these magazines to be put in individual, sealed modesty bags.
"The publishers of these magazines now have until 9 September to start providing their own modesty bags, after which any lads' magazine which does not have the relevant bag will not be supplied in our stores."
The company's retail arm, The Co-operative Food, claims to be one of the biggest magazine retailers in the country, with more than 4,000 stores.
A spokeswoman for Bauer Media's Zoo magazine did not confirm whether the title would use modesty bags, but said: "Bauer Media is a responsible publisher and supports the existing best-practice guidelines for the display of men's magazines.
"We are sensitive to the mood of the public and to that end we have responded accordingly, and have changed Zoo magazine's cover imagery and phrasing.
"We already have agreements in place with all major retailers, including the Co-op, to ensure Zoo magazine is displayed appropriately and we work closely with all our retailers to ensure they are adhered to."
A spokesman for the PPA said: "The average age of a reader of men's lifestyle magazines is 30, according to the National Readership Survey, and these titles are not created for, or marketed to, children.
"Publishers support the guidelines on the appropriate display of men's lifestyle magazines, which have been drawn up with the National Federation of Retail Newsagents and endorsed by the Home Office."