BRUSSELS (Reuters) - The European Union could send trainers to Libya in the coming months to help the new government secure borders against weapons smuggling, under a proposal discussed on Monday.

At a meeting in Brussels, EU foreign ministers considered plans for a mission to try to stop the trafficking of arms, which have flowed out of Libya since the overthrow of Muammar Gaddafi last year, reached al Qaeda militants and fanned a separatist rebellion in northern Mali.

European governments are also worried about illegal migration. Under Gaddafi, Libya was a major route for migrants from Africa to Europe until 2009, when Italy signed an agreement with Libya to curtail it.

Italy pressed its EU counterparts on Monday to move quickly to establish a border guard training mission with Libya.

"The Italians said they wanted a mission in January. That's a bit unrealistic," said one EU diplomat.

EU officials said trainers could arrive in Libya by mid-year, helping the government develop land and sea border security systems. Italy has already stepped up cooperation with the new Libyan authorities this year to improve monitoring and border controls.

According to Frontex, the EU's border agency, some 3,500 migrants have arrived on European shores, mainly in Italy and Malta, from Libya in the first 10 months of this year.

(Reporting by Justyna Pawlak; editing by Andrew Roche)