By Michelle Martin and Humeyra Pamuk

BERLIN/ISTANBUL (Reuters) - Turkey could walk away from its promise to stem the flow of illegal migrants to Europe if the European Union fails to grant Turks visa-free travel to the bloc in October, Foreign Minister Mevlut Cavusoglu told a German newspaper.

His comments in Bild's Monday edition coincide with rising tensions between Ankara and the West that have been exacerbated by the failed coup attempt in Turkey on July 15. Turkey is incensed by what it sees as an insensitive response from Western allies to the failed putsch, in which 240 people were killed.

Long wary of Turkey's ambitions to join the EU, Europe has been alarmed by the crackdown since the coup, fearing President Tayyip Erdogan is using purges to quash dissent. The unease has relations between Turkey and Austria and Sweden. Ankara has summoned diplomats from both countries to protest what it says are false reports about changes to its child abuse laws.

Asked whether hundreds of thousands of refugees in Turkey would head to Europe if the EU did not grant Turks visa freedom from October, Cavusoglu told Bild: "I don't want to talk about the worst case scenario - talks with the EU are continuing but it's clear that we either apply all treaties at the same time or we put them all aside."

Visa-free access to the EU - the main reward for Ankara's collaboration in choking off an influx of migrants into Europe - has been subject to delays due to a dispute over Turkish anti-terrorism legislation, as well as the post-coup crackdown.

Brussels wants Turkey to soften the anti-terrorism law.Ankara says it cannot do so, given multiple security threats which include Islamic State militants in neighboring Syria and Kurdish militants in Turkey's mainly Kurdish southeast.

European Commissioner Guenther Oettinger has said he does not see the EU granting Turks visa-free travel this year due to Ankara's crackdown, which has included the round-up of more than 35,000 over alleged involvement in the coup.

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Cavusoglu said the migration deal with the EU stipulated that all Turks would get visa freedom in October, adding: "It can't be that we implement everything that is good for the EU but that Turkey gets nothing in return."

A spokesman for the European Commission declined to comment on the interview directly but said the EU continued to work together with Turkey in all areas of cooperation.

SWEDEN, AUSTRIA

Since the coup, more than 17,000 people have been placed under formal arrest, and tens of thousands more suspended from their jobs. Turkish authorities blame the failed putsch on U.S.-based Muslim cleric Fethullah Gulen and his followers. Gulen denies involvement and has condemned the coup attempt.

Cavusoglu told reporters that the Ankara government had summoned Sweden's ambassador to protest at comments from Foreign Minister Margot Wallstrom, a day after it hauled in Austria's charge d'affaires.

"(The) Turkish decision to allow sex with children under 15 must be reversed. Children need more protection, not less, against violence, sex abuse," Wallstrom wrote on her official Twitter account.

Cavusoglu dismissed her comments as "the result of racism and anti-Islam sentiment in Europe ... It is a scandal for a foreign minister to tweet something like this based on false rumors. It is worrying that this campaign of lies, which started in Austria, has spread to Sweden."

Her comments were in reference to a decision by Turkey's constitutional court last month to remove a provision in the penal code which identifies all sexual acts against children under the age of 15 as "sexual abuse".

A Turkish official said the claim that sexual abuse of children under 15 would now go unpunished was "completely baseless" and that new legislation would go into effect before the court ruling does to plug any legal loopholes.

On Sunday, Turkey summoned Austria's charge d'affaires to protest at a headline on an electronic news ticker at the airport in Vienna that allegedly read, "Turkey allows sex with children under the age of 15". Austrian officials played down the matter as one of freedom of the press.

A Turkish Foreign Ministry statement said the publication of such "slandering" news reports were encouraged by recent comments from Austrian politicians.

Austrian Chancellor Christian Kern has said the EU should end accession talks with Turkey, prompting Cavusoglu to refer to Austria as the "capital of radical racism". On Monday, Kern said that if the EU broke off accession talks with Turkey, this should not impact the migrant deal.

"If this deal depends on the illusion of accession talks, then we have a big problem," Kern told Austrian news agency APA.

(Additional reporting by Julia Fioretti in Brussels, Tuvan Gumrukcu in Istanbul and Francois Murphy in Vienna; writing by David Dolan; Editing by Paul Carrel and Richard Balmforth)