The European Commission will grant Turkey conditional approval for visa-free travel within the EU's Schengen area, sources have told the BBC.
The move is part of a deal in which Turkey has agreed to take back migrants who have crossed the Aegean to Greece.
But Turkey must still meet EU criteria, BBC Europe Editor Katya Adler says, and the deal must be approved by the European Parliament and member states.
The EU fears that without a visa deal, Turkey will not control migration.
The huge numbers of migrants and refugees arriving in Europe from Turkey, and from North Africa, has caused a political crisis among EU states.
A formal announcement from the European Commission is due on Wednesday.