* At least six refiners to receive full requested volumes in June

* One buyer gets a small cut

* Some refiners asked for less Saudi crude because of high prices

SINGAPORE/NEW DELHI, May 11 (Reuters) - Saudi Arabia, the world's top crude oil exporter, will supply full volumes of crude to most Asian refiners in June, several people with knowledge of the matter said on Tuesday.

The OPEC kingpin started easing supply cuts to buyers in May as OPEC, Russia and their allies, a group known as OPEC+, stuck to plans for a phased roll-back of oil production restrictions from May to July.

Indian state refiners requested regular supplies from Saudi state energy giant Saudi Aramco for June, after reducing purchases this month.

One of the sources said most Asian refiners received their allocations while Saudi Aramco has cut supplies to Europe.

Still, Saudi Aramco is trimming supplies to at least one Asian buyer in June within a permissible adjustment limit in the contract, the sources told Reuters.

Under the contracts, the seller or the buyer can adjust loading volumes, depending on demand and shipping logistics, using operational tolerance which ranges from plus to minus 10% of the contracted Saudi volume.

Saudi Aramco declined to comment.

Despite Saudi Aramco cutting prices for its Asian supplies in June for the first time in six months, some buyers had requested lower volumes as the price of flagship Arab Light crude was relatively higher than similar grades in the spot market such as Upper Zakum from Abu Dhabi, the sources said.

"Upper Zakum looks much cheaper than Arab Light," one of the sources said.

Falling fuel demand in countries such as India and Japan because of COVID-19 lockdowns could have also reduced those refiners' appetite for more crude, the sources said.

(Reporting by Florence Tan and Nidhi Verma; Editing by Muralikumar Anantharaman, Richard Pullin, Louise Heavens and Susan Fenton)