BRUSSELS (Reuters) - NATO will assume a greater role in the coordination of arms supplies to Ukraine, the alliance said on Friday, taking over from the U.S. in a bid to safeguard the aid mechanism as NATO-sceptic Donald Trump bids for a second term as U.S. president.

"These efforts do not make NATO a party to the conflict but they will enhance our support to Ukraine to uphold its right to self-defence," NATO Secretary-General Jens Stoltenberg told reporters in Brussels.

On Wednesday, Hungary had given up its resistance to the Ukraine support package NATO aims to agree at its Washington summit in July, comprising a financial pledge and the transfer to NATO of the coordination of arms supplies and training.

During a visit by Stoltenberg to Budapest, Prime Minister Viktor Orban said his country would not block NATO decisions on providing support for Ukraine but had agreed that it would not be involved.

Hungary has been at odds with other NATO countries over Orban's continued cultivation of close ties to Russia and refusal to send arms to Ukraine.

After Russia's full-scale invasion of Ukraine in 2022, the United States gathered like-minded nations at the Ramstein air base in Germany, establishing a group of now some 50 nations that meet regularly to match Kyiv's arms requests with pledges of donors.

This so-called Ramstein group will continue to exist as a U.S.-led political forum but NATO will take over the military working level below that coordinates arms deliveries and training for Ukrainian troops.

The move is widely seen as an effort to provide a degree of "Trump-proofing" by putting coordination under a NATO umbrella, giving the alliance a more direct role in the war against Russia's invasion while stopping well short of committing its own forces.

But diplomats acknowledge such a move may have limited effect, as the U.S. is NATO's dominant power and provides the majority of weaponry to Ukraine. So if Washington wanted to slash Western aid to Kyiv, it would still be able to do so.

(Reporting by Sabine Siebold; Editing by Benoit Van Overstraeten)