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The European Commission headquarters is seen in Brussels, Belgium March 17, 2016. REUTERS/Yves Herman

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By Gabriela Baczynska and Paul Taylor

BRUSSELS (Reuters) - The European Union is set to miss an end-June deadline to grant visa-free travel to Turkey because of a dispute over Turkish anti-terrorism law, barring an improbable sudden concession by President Tayyip Erdogan, EU officials and diplomats say.

The main reward for Ankara's collaboration in choking off an influx of migrants into Europe may now come at the earliest in July but more likely not until the autumn, they say.

The 28-nation bloc is dependent on Ankara to enforce a deal, criticised by rights groups, that has sharply cut the number of refugees and migrants reaching Greece, giving EU leaders breathing space after more than a million arrived last year.

However, there is no sign for now of the deal falling apart.

Allowing Turks to visit Europe's 26-nation Schengen area visa-free for up to three months is unpopular in many EU states, including France, where authorities fear it could be a gift to anti-immigration populist Marine Le Pen in elections next year.

Ankara has yet to meet five of the 72 requirements for visa liberalisation, according to the executive European Commission, which proposed this month relaxing travel rules for Turks if Turkey fulfilled those benchmarks by the end of June.

But the European Parliament, which must approve the visa decision, has refused to start work on the proposal until Ankara meets the criteria in full. Parliament President Martin Schulz said he did not see that happening before the end of June.

"De facto a decision in June is not possible anymore," said an EU source familiar with the negotiations. Lawmakers could still theoretically vote on the plan at their final plenary session on July 4-7, but more likely after the summer break.

Turkish Prime Minister Ahmet Davutoglu, who won the June target date as part of the wider migration deal in March, is being replaced after losing a power struggle with Erdogan, who has refused to amend anti-terror laws to meet the EU criteria.

Erdogan's main priority is to change the constitution to secure extensive presidential rule and he has said October would also be acceptable for visa liberalisation.

"An October scenario is possible, but also July. Let's see how the Turks comply," another EU source said.

Meanwhile, Turkey continues to stop migrants leaving its shores and both sides have a vital interest in keeping the deal alive.

EU SEES BALL IN TURKEY'S COURT

The EU says Turkey must narrow its definition of terrorist crimes, which leads to extensive application of the law against intellectuals, Kurdish sympathisers and critics of Erdogan, including dozens of journalists and academics.

Turkey has repeatedly declined to do so, saying the law is crucial to its fight with Kurdish and Islamic State militants.

The EU is now waiting to see what a new government will do on the outstanding criteria after a meeting of the ruling AK Party on Sunday picks a new prime minister, diplomats said.

"We've been through this before," said another EU official with long experience of negotiating with Ankara. "It's not the first time there has been quite provocative talk from the Turkish side, then we sat down and found a way forward."

Sources in Brussels stress that the Commission's report said Turkey should "better align" - rather than "fully align" - its counter-terrorism regulation with EU norms, including to bring in more proportionality in punishment for any violations. That offers room for compromise.

Keeping the migration accord on track is a key priority for several EU member states, especially the bloc's biggest power, Germany, which took in most of the 1.3 million refugees and migrants who reached Europe last year.

The EU also promised revitalised EU accession talks and an initial 3 billion euros in aid for Syrian refugees in Turkey in return for Ankara's help on migration, and it has taken steps recently to speed up disbursement.

Turkey is only due to start issuing the advanced biometric passports required for visa-free travel in October. No Turks yet have the new generation of passports, EU officials say.

Given the European Parliament's tough stance on human rights and public anxiety in several EU states about granting visa-free entry to a largely Muslim population of 79 million, the room for compromise is limited.

"We've got to get something that is more than a symbolic gesture from them on terrorism. The market will demand it," the first EU source said.

(Writing by Gabriela Baczynska; Editing by Paul Taylor and Gareth Jones)

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