The European Commission will back visa-free travel for Turkish citizens inside Europe’s passport-free Schengen area, sources have told the BBC.
Visa liberalisation was offered in return for Turkey taking back migrants who crossed the Aegean Sea to Greece.
But Turkey must still meet EU criteria, and the visa deal needs approval by the European Parliament and member states.
The EU fears that without this deal, Turkey will not control migration.
The waiver would scrap the requirement for Turks to get a three-month, short-stay Schengen visa, for tourism or business trips. But it will not grant Turks the right to get a job in Europe.
The UK, Ireland and Cyprus are not in Schengen, so they will keep the visa requirement for Turkey.
Turkey has threatened to stop taking back migrants from Greece if the EU fails to deliver on visa liberalisation.
A long-running dispute with Cyprus is a major stumbling block in Turkey’s bid to join the EU, not least because Ankara does not recognise the Republic of Cyprus, an EU member.
Turkish-controlled northern Cyprus does not have international recognition.
But a Turkish official, quoted by Reuters news agency, said a visa deal with the EU would also mean Turkey scrapping its visa requirement for Greek Cypriots.
The large influx of migrants and refugees arriving in Europe from Turkey, and from North Africa, has caused a political crisis among EU states.