A spokesman for the council said that talks with Associated had been productive and an announcement is expected for mid-next week.
"There is a lot at stake for both the publishers and ourselves," he said.
The council issued a statement last week stating that Associated's London Lite and News International's thelondonpaper would be banned if they did not significantly contribute to the £500,000 recycling plan that it had drawn up to deal with the extra waste created by the free papers.
The council had increased the number of newspaper recycling bins to 131 following the launch of the free evening newspapers last August, but it said it needed to introduce an extra 300 bins and extra lorries, with crews to regularly empty them, because the freesheets are creating an extra three to four tonnes of waste in the borough each day.
Local authorities have the power under the Clean Neighbourhoods and Environment Act 2005 to restrict the distribution of free literature in areas where they pollute the environment. No other council in London has threatened to use this law to date.
The freesheets were also in trouble this week for allegations of newspapers dumping, which the Audit Bureau of Circulation said it will now investigate.
Associated's Evening Standard released video evidence of three vendors from thelondpaper dumping 2,900 spare copies in bins near Barbican, St Paul's and Liverpool St, on three separate occasions in March.
This led thelondonpaper, published by News International, to launch an advertising campaign on Wednesday, which said that it had sacked the three vendors involved.
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