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.
Migration pressure - analysis by Katya Adler, BBC Europe Editor
If the European Commission (the EU's executive body) does make the recommendation on Wednesday that Turks be granted visa-free travel in Europe's Schengen area, as whispers from well-placed EU sources suggest, then it will be doing so holding its nose and its breath.
Freedom of speech; the right to a fair trial; revising terrorism legislation to better protect minority rights - these are just some of the criteria demanded by the EU of countries before it lifts visa requirements, even for short-term travel.
It is hard to see how Turkey could be described as meeting these conditions. The government in Ankara increasingly cracks down on its critics in a manner more autocratic than democratic.
But these are desperate times for the EU. The European Commission and most EU governments are under huge public pressure to ease the migrant crisis.
My sources say the Commission will therefore keep to the agreed script. But they insist this is no blank cheque. Turkey will get the green light over visas this week to keep it sweet. But it will also be informed of the outstanding criteria it still needs to meet.
Is the visa waiver likely to happen?
There is a good chance it will. EU politicians are under huge pressure to keep sending migrants back to Turkey, even though the deal is very controversial. And Turkey insists that the EU must fulfil its offer of visa liberalisation by the end of June.
However, British MEP Claude Moraes told the BBC that the deal would come under very tough scrutiny in the European Parliament.
Many Turks regard the visa requirement as humiliating - some have missed study opportunities or business conferences in the EU for want of a visa. And scrapping visas would signal that the EU is serious about Turkey's negotiations to join the bloc.
The EU is already moving towards granting visa-free travel to nationals of Kosovo, Georgia and Ukraine. That makes it harder to justify leaving Turkey out - especially as those countries are not yet candidates for EU membership, unlike Turkey.
Under the EU-Turkey agreement, migrants who have arrived illegally in Greece since 20 March are to be sent back to Turkey if they do not apply for asylum or if their claim is rejected.
For each Syrian migrant returned to Turkey, the EU is to take in another Syrian who has made a legitimate request.
Under the agreement, Turkey must meet 72 conditions by 4 May to earn visa-free access to the Schengen area by the end of June. Diplomats have suggested that fewer than 10 still need to be met.
Human rights groups question the deal's legality and argue that Turkey is not a safe place to return people to.
Last month, however, European Council President Donald Tusk said the deal had begun to produce results.
He praised the Turkish government as "the best example in the world on how to treat refugees".
What are the arguments for and against?
Concerns have been raised in the European Parliament that this looks like a reward for Turkey, because of its co-operation in the migrant crisis. Ankara falls short of many EU human rights benchmarks.
MEPs accept that Turkey is a "key strategic partner" for the EU. But they say reforms have slowed down in Turkey in many areas, including freedom of speech and judicial independence.
MEPs voiced concern about continuing fighting in south-eastern Turkey between government troops and Kurdistan Workers' Party (PKK) rebels, whom the government in Ankara describes as "terrorists".
The European Commission's 2015 report on Turkey also complained of politicisation of the Turkish judiciary, widespread corruption, inadequate protection of minority rights and "significant backsliding" on freedom of speech and assembly.
On the plus side, however, the Commission says Turkey has improved conditions for the many Syrian refugees it is hosting - some 2.75 million. It praises Turkey for providing Syrians with access to jobs and schooling for their children.
It also says Turkey is tightening up its admission rules for migrants from countries which "are sources of significant onward migration" to the EU. Afghanistan, Iran and Pakistan are among those countries.
Many nationalities can already visit Europe's Schengen zone visa-free, including nationals from Latin America and the Caribbean, Hong Kong and South Korea.
A note on terminology: The BBC uses the term migrant to refer to all people on the move who have yet to complete the legal process of claiming asylum. This group includes people fleeing war-torn countries such as Syria, who are likely to be granted refugee status, as well as people who are seeking jobs and better lives, who governments are likely to rule are economic migrants.