The France international, who had a 12-month contract extension triggered in April, is expected to sign a two-year deal with the Serie A giants.
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Milan had wanted to sign the 34-year-old on a free, but will pay around £850,000 with a further £850,000 in potential add-ons.
Chelsea have already sold Fikayo Tomori to the Italian club for a fee that could rise to £30million and are set to bring in more funds by offloading a host of fringe players during this window.
They want £40million for striker Tammy Abraham, while left-back Emerson Palmieri is wanted by Napoli among other Italian clubs.
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