* Motivated in part by fears of new migrant wave from Libya

* EU mulling police training but no military presence

* Separate naval mission may target migrant trafficker boats

* U.S. keen to see Europe take lead in Libya, diplomats say

By Robin Emmott

BRUSSELS, April 15 (Reuters) - The European Union is signalling that it will consider moving security personnel into Libya to help stabilise the chaotic country if requested by a new U.N.-backed Libyan government, according to a draft statement seen by Reuters.

Impetus for the move comes in part from fears of an uncontrolled new tide of migrants into Italy from Libya unless law and order can be rebuilt soon in the North African state.

EU foreign and defence ministers will hold a special dinner on Monday in Luxembourg where they are expected to agree to look into police and border training missions for Libya, initially in Tripoli, where the new government is trying to establish itself.

"The EU stands ready to offer security sector support in response to possible (U.N.) Government of National Accord requests," ministers are expected to say, according to a draft statement prepared by diplomats that is still under discussion.

"A possible civilian ... mission could support Libyan efforts ... through advice and capacity building in the fields of police and criminal justice," said the draft, referring to counter-terrorism, border management and tackling the smuggling of migrants across the Mediterranean to Europe.

An EU security presence in Libya, which would not involve soldiers, would be Europe's biggest step in the oil-producing nation since a NATO-backed mission led to the fall of Libya's long-time leader Muammar Gaddafi in 2011.

Diplomats said there had yet to be a detailed discussion with the new U.N.-brokered Libyan government in defining what kind of assistance they wanted from the EU, and that it keen to avoid the impression of moving into the country uninvited.

"It is a delicate balance," said one senior EU official involved in the plans. "We need to prepare to help Libya, but we cannot jump the gun."

Libyan officials with the new unity government were not immediately available for comment on the specific document. But they have said any international security cooperation must be Libyan-led and so far have made no detailed request for aid.

But inviting in foreign military trainers remains a sensitive subject for the new government, who opponents accuse of being a foreign-imposed body with no legitimacy.

LIBYAN PM TO ADDRESS EU MEETING

Libyan Prime Minister Fayez Seraj will speak to ministers by video conference at the dinner on Monday.

Talks on a possible EU security mission could give momentum to discussions among Italian, French, British and U.S. military planners on whether to send troops to Libya to help protect key installations, government buildings, ports and the airport.

The United States is eager to see Europe, not Washington, take the lead in a region on the continent's doorstep.

The separate mission, which includes France, Italy, Britain and the United States and is known as the Libya International Assistance Mission, has already briefed EU diplomats about how it could have a military role in stabilising Libya. It may set up a secretariat based in Rome.

Also under consideration is how the EU's so-called "Sophia" naval mission operating in international waters near Libya could move into Libyan waters to destroy boats used by people smugglers, catch the traffickers and head off an expected surge in migrants trying to reach Europe by sea from Libya.

While the naval mission has been operating since mid-2015 and has saved more than 8,000 lives, it is unable to move into Libyan waters without a request from the Libyan government and a U.N. Security Council resolution.

The problem has been finding an effective governing authority in Libya to deal with. Libya has been in anarchy for years, with two competing governments based in Tripoli and the far east and a plethora of militias dominating various regions. The new U.N.-backed unity government has yet to stamp its authority in Tripoli, let alone the vast country at large.

Previous training efforts ran into difficulties in 2012 and 2013 when Italy and Turkey started training police and, along with Britain and the United States, planned to build up a force of 8,000 troops. Those programmes were hampered by militia infighting and political squabbling among Libyan factions.

EU ministers will also discuss whether Sophia can share information and work more closely with NATO's naval mission in the Aegean, which aims to help Greek and Turkish coastguards tackle people smuggling there and stem the record flows of migrants into the EU via Greece from nearby Turkey. (Additional reporting by Patrick Markey; Editing by Francesco Guarascio and Mark Heinrich)