* Alliance to discuss 'arc of instability' to the south and east

* EU foreign affairs chief to brief on migrant crisis

* NATO considers Iraqi training request (Adds Turkish foreign minister)

By Adrian Croft and Tulay Karadeniz

ANTALYA, Turkey, May 12 (Reuters) - Preoccupied for more than a year by the Ukraine crisis, NATO foreign ministers meeting in Turkey this week will focus on instability on the alliance's southern flank, ranging from Islamic State in Iraq and Syria to turmoil in Libya.

By meeting in Turkey, which shares a 1,200 km (750-mile) border with Iraq and Syria, NATO hopes to show it is responsive to the concerns of its southern members, as well as reinforcing eastern European allies worried by Russia's actions in Ukraine.

"Turkey is the only country which neighbours areas under Islamic State control in Syria and Iraq," Turkish Foreign Minister Mevlut Cavusoglu told a news conference in the southern city of Antalya before the meetings on Wednesday and Thursday.

"This is not sustainable, and a big threat to Turkey. The summit will be an opportunity to share these views."

U.S. Ambassador to NATO Douglas Lute described an "arc of instability" around the east and south of the alliance, with the "maybe failed state of Libya" a funnel for illegal immigration from states such as northern Nigeria, Mali, Niger and Somalia.

"You've got, to the east, to the southeast and to the south, pretty fundamental security challenges for NATO," he told reporters at a briefing late last week.

Libya has descended into lawlessness since rebels overthrew strongman Muammar Gaddafi in 2011 with the help of a NATO bombing campaign.

NATO as an organisation has not been highly active militarily in the south recently, although all of its member nations are part of the U.S.-led coalition against Islamic State. It also sent Patriot anti-missile systems to defend Turkey from possible attack from war-torn Syria.

The alliance is considering a request from Iraq for help training its army. But it says the security situation in Libya must improve before it could help train Libyan security forces.

EU foreign policy chief Federica Mogherini is expected to brief the NATO ministers on the EU's proposals for a military mission to capture and destroy smugglers' boats used to ferry migrants on perilous Mediterranean crossings from Libya.

NATO Secretary General Jens Stoltenberg said NATO had not been asked to play a military role in the mission.

But Lute said information-sharing between NATO and the EU could be possible. Accurate intelligence pinpointing smugglers' vessels would be key to the success of the operation.

Ministers will discuss the situation in Ukraine and will also talk about whether NATO should keep a presence in Afghanistan after the current NATO-led training mission ends, probably at the end of 2016.

The most likely outcome is that NATO will then retain a small civilian-led mission, including both soldiers and civilians, a NATO source said. (Editing by Nick Tattersall and Mark Trevelyan)