By Crispian Balmer

ROME, April 21 (Reuters) - Italian officials played down on Friday suggestions that U.S. President Donald Trump had snubbed efforts by Prime Minister Paolo Gentiloni to get Washington more involved in bringing political stability to Libya.

Gentiloni met Trump at the White House on Thursday, telling reporters he was seeking support to help deal with the North African state that fragmented and descended into chaos after the downfall of Muammar Gaddafi in 2011.

"The U.S. role in this is very critical," Gentiloni told a news conference standing alongside Trump. The U.S. president immediately responded: "I do not see a role in Libya. I think the United States has right now enough roles."

Italian commentators said Trump appeared to have brushed aside Gentiloni's request for help. "Libya divides Trump and Gentiloni," the pro-government la Repubblica newspaper said in a headline. "The alliance in difficulty," it said.

However, reporters following the news conference said Trump had not been listening to the simultaneous translation when Gentiloni had made his remark and an official close to the Italian prime minister said the meeting had gone well.

"There was no snub at all," said an official close to Gentiloni, adding that two-thirds of the leaders' talks on Thursday had centred on Libya.

As proof of the U.S. involvement in Libya, Italian officials said Washington had invited Fayez Seraj, who heads a U.N.-backed government in Tripoli, and Khalifa Haftar, a prominent commander in eastern Libya, for talks in the U.S. capital later this year.

There was no immediate confirmation from Washington.

Haftar, who is backed by Russia and Egypt, is a figurehead for east Libyan factions. Italy has thrown its support instead behind Seraj in the west and warns that the divisions could split Libya apart, with dire consequences for the whole region.

Rome is especially anxious to restore peace because Libya has become a hub for human traffickers, with more than half-a-million migrants reaching Italy over the past three years on a flotilla of rickety ships and inflatable boats.

"Now is the moment for the U.S. and Italy to work together to stabilise the situation and broaden the support for the Tripoli government to other actors," Gentiloni said in a speech in Washington on Thursday before he saw Trump.

"The division of Libya is not a good idea. It would be dangerous for Egypt, dangerous for Tunisia and for the interests of Europe," he told the Center for Strategic and International Studies (CSIS).

He added that NATO countries including the United States had an especial responsibility to Libya, saying their 2011 military intervention had got rid of Gaddafi but had ushered in mayhem.

"It was evidently an intervention that lacked a vision or perspective for the future," Gentiloni said.

Italy is worried that the United States might eventually throw its weight behind Haftar at the request of Egypt, which promotes the general as a bulwark against Islamist militants.

(Reporting by Crispian Balmer; Editing by Tom Heneghan) ((crispian.balmer@thomsonreuters.com; +390685224351; Reuters Messaging: crispian.balmer.reuters.com@reuters.net))