* More attacks similar to Jewish Museum shooting likely
* A third of Europeans fighting in Syria may have returned
* ISIL advance in Iraq may encourage jihadists to go there
By Adrian Croft
BRUSSELS, June 19 (Reuters) - The hardline Islamist group that has seized parts of Syria and Iraq is probably training and directing foreign fighters to carry out attacks in Europe and elsewhere, the European Union's counter-terrorism coordinator said on Thursday.
Gilles de Kerchove also said it was likely that there would be more attacks in Europe like the shooting at the Jewish Museum in Brussels last month that killed four people.
Mehdi Nemmouche, a 29-year-old Frenchman believed to have returned recently from fighting with Islamist militant rebels in Syria, was arrested in May for the Jewish Museum killings.
At the time of his arrest, Nemmouche had a Kalashnikov wrapped in a flag with the inscriptions of the Islamic State of Iraq and the Levant (ISIL), an al Qaeda offshoot involved in Syria's civil war and which also spearheaded last week's offensive across north and western Iraq.
Investigation would show whether the Brussels shooting was part of a planned strategy by ISIL, also known as ISIS, de Kerchove said in an interview with Reuters.
There was a question about whether the suspect in the Brussels shooting acted alone or under direction, he said.
But he said it was "very likely that the ISIS ... maybe is preparing, training, directing some of the foreign fighters to mount attacks in Europe, or outside Europe."
ISIS shared with al Qaeda the idea of a global caliphate "so it makes sense that they send fighters back and mount attacks outside the region," he said.
British Prime Minister David Cameron said on Wednesday that Sunni Islamist insurgents fighting in Iraq were planning to attack Britain and that "ungoverned spaces" where militants thrived had to be shut down.
Asked if he expected more attacks in Europe similar to the attack at the Jewish Museum, de Kerchove, a Belgian, said: "I am worried and I think it is likely."
He said he was more afraid of a repeat of an attack similar to the Brussels shooting in Europe than he was about the possibility of a large-scale attack like the 2001 plane attacks on U.S. cities. "I think we are much smarter than we were on 9/11 to prevent that big-scale attack," he said.
The Brussels attack has raised alarm about the risk of Europeans going to fight in Syria and staging attacks in Europe when they return.
De Kerchove estimated that more than 2,000 Europeans had gone to fight in Syria.
"A third may have already come back but most of them, the bulk, remains there," he said.
Advances made by ISIL fighters in Iraq may give an extra incentive for would-be jihadists to go there because "for those who are attracted by the idea of the caliphate ... it may be exciting to join because that shows successes," he said.
He said Europe had made a lot of progress over the last 18 months in putting in place measures to prevent people travelling to Syria, detecting people who had been there and sharing information among European countries.
It was beyond the resources of European security services to monitor around the clock all the people returning from Syria, he said. "We have to design a mechanism to assess the dangerousness of each and every returnee," he said.
Earlier this month, nine European countries endorsed plans to step up intelligence-sharing and take down radical websites to try to stop European citizens going to fight in Syria and bringing violence back home with them.
De Kerchove, calling Syria "the first Twitter war", said EU officials had held talks with internet or social media giants such as Google, Twitter and Facebook to discuss what more could be done to tackle radicalisation.
Ideas were still being worked on but the initiative was about using the internet to offer a counter-narrative to radical messages as well as being more effective in removing illegal web sites, he said. (Editing by Ralph Boulton)