BRUSSELS—European Union countries will have to take large numbers of refugees directly from Turkey as part of a longer-term strategy to deal with the migration crisis, said Frans Timmermans, the European official who has been leading negotiations on the issue with the Turkish government.

The leaders of the European Union and Turkey agreed Sunday that Ankara would take steps to cut the flow of migrants into Europe in exchange for EU cash and help with its bid to join the 28-nation bloc.

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