The newspapers will carry out the regular litter collections, in addition to the council's existing street sweeping, which will aim to reduce the paper waste in the zones of Charing Cross and Embankment; Leicester Square and Charing Cross Road; Oxford Circus; and Victoria station.
The publishers will be allowed to brand the bins with their respective logos once they receive advertising consent.
Councillor Alan Bradley, cabinet member for street environment said: "I'm very pleased that both publishers have agreed to help tackle the problem of newspaper litter voluntarily, which was always our favoured option.
"This has been a complex matter, and there are some details we need to finalise, but I look forward to all parties working together to ensure Westminster's streets are kept clean and that as much waste newspaper as possible is recycled. I'm confident we'll see the benefit of all the hard work put into this once the relevant planning and advertising consents have been obtained and new bins delivered."
The deal comes after months of negotiating between the freesheets and the council, which threatened to "severely curtail" distribution of both papers back in April if their publishers did not significantly contribute to its £500,000 recycling plan, designed to deal with the extra four tonnes of waste created by the papers every day.
The council's recycling plan included buying an extra 300 bins and extra lorries for the West End, with crews to regularly empty them, at a total cost of £500,000.
The freesheets hired their own teams to clear-up and take away the newspapers left in the streets, but by July, after months of negotiations, they had still not reached a financial agreement with Westminster City Council for its recycling plan.
The council set an August 9 deadline, by which time an agreement had to be made or it would invoke the Clean Neighbourhoods and Environment Act 2005.
Under this act, the freesheets would have been required to hold a permit to distribute their papers. However, not only the freesheets would have been affected, any individual or organisation distributing any free literature in the West End would also have required a permit.
Steve Auckland, managing director of London Lite and Metro, said the deal had been a long time coming: "We spoke to the councils last October and we said there could be issues [with recycling].
"We're obviously trying to work responsibly -- we're not putting too many copies out, and we want to get our merchandisers off the street by 7.30pm."
Ian Clark, general manager of thelondonpaper, said: "Thelondonpaper has worked hard with Westminster Council over the last few months to resolve the issues surrounding the litter from free newspapers.
"Although the London Lite has grudgingly matched our voluntary agreement with Westminster, Lite's record in this area is less than impressive when compared to thelondonpaper. It is disappointing that Associated's other titles, The Evening Standard and Metro, have declined to play their part in this voluntary scheme."
Thelondonpaper distributes just over 500,000 copies each day, while London Lite hands out just over 400,000, according to the latest ABC figures.
The new regime will take effect later this year at a date agreed by all parties and is scheduled to last for an initial three-month period, before a full review is carried out ahead of further negotiations to extend it.