MOSCOW, June 1 (Reuters) - Urals crude differentials remained stable to dated Brent on Monday in a quiet market, while the EU may keep the price cap for Russian oil at a low $44 per barrel despite higher global prices, in a push to complicate trading and shipping for the volumes, EU diplomats said.

The Group of Seven nations and allies, with the exception of the U.S., agreed to set a moving price cap last year to make it more effective. The countries cut the price from $60 per barrel to $47.60 to reflect lower average oil prices and in January, they revised the price down to $44.10.

Though global oil prices and thus the Urals price have risen significantly this year, the EU plans to keep the price cap level at $44 per barrel in an effort to curb Moscow's windfall from the Iran war and the ensuing oil price shock.

The Commission may also propose that any future price review cannot top $60 per barrel regardless of average prices at the time, sources said, given the outlook of sustained high prices for the remainder of the year.

Under the price cap mechanism, third countries can buy Russian oil up to that maximum price using Western shipping and insurance services. Up to 20% of seaborne Russian oil is still traded under the cap, while the rest is moved by a shadow fleet.

June Urals oil cargoes continue to trade with most volumes destined for Indian ports, traders said.

PLATTS WINDOW

• There were no bids or offers on Friday for Urals, CPC Blend and Azeri BTC, traders said.

NEWS

• Kazakhstan has restored its oil production to 290,000 metric tons per day following earlier production losses at the country's largest oilfield, Tengiz, Energy Minister Erlan Akkenzhenov said on Monday.

• U.S. crude exports climbed to a record 5.6 million barrels per day in May as the Middle East crisis pushed up demand for the country's oil from Asian and European refiners, ship tracking estimates showed on Monday.