The Capital Pub Company, which has 34 pubs, has accepted the £70m cash offer.
It rejected Fullers' £54m takeover approach last month dismissing it as insufficient.
Greene King wants to expand its presence in what it described as the "attractive and growing premium eating and drinking out market in London".
If the deal is accepted by Capital's shareholders, the combined group will have 250 pubs in Greater London.
Greene King's offer is for 235 pence per share. Capital's shares were trading at 155.5p before the offers were made. Fuller's rejected offer was for 200p.