The publishers have been in separate negotiations with Westminster City Council since it decided that, if they could not reach a voluntary agreement to limit the litter caused by their papers, their distribution would be forcibly curtailed.
Both publishers have until 9 August to reach an agreement.
Media Week understands that each publisher is close to agreeing to fund 40 specially made recycling bins and to pay for the collection and recycling of the waste papers.
The bins would be branded with London Lite's logo on one side and thelondonpaper's on the other, as well as Westminster Council's logo.
The bins would be supported by messages in the papers asking readers to dispose of them carefully. They will be placed on litter hotspots identified by the council.
Doug Read, Associated executive director of free newspapers, said: "The message is that we are keen to encourage people to recycle."
News International confirmed the negotiations were happening, but was unable to comment further at the time of going to press.
A Westminster Council spokesman confirmed the arrangements were being finalised, adding: "Any prospect of the statutory regime now seems extremely unlikely."
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