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* Prices mixed after big gains last week
* U.S. imposes sanctions on Iranian leadership
* Interactive graphic on oil sanctions: https://tmsnrt.rs/2FA66mz (Changes dateline to London, updates prices, adds detail, analyst comment)
LONDON, June 25 (Reuters) - Oil prices hung in the balance on Tuesday as concerns over declining crude demand were offset by risks to supply linked to new U.S. sanctions on Iran.
Benchmark Brent crude futures were down 12 cents at $64.74 a barrel by 0832 GMT.
U.S. crude futures were up 3 cents at $57.93 a barrel.
Hopes for progress in the trade war between China and the United States during this week's G20 meeting were dampened by a comment from a senior U.S. official saying U.S. President Donald Trump was "comfortable with any outcome" from the talks.
Weak manufacturing data released on Monday by the Federal Reserve Bank of Dallas added to worries about slipping demand for crude oil due to the trade conflict.
"The geopolitical risk premium (in the Middle East) is partly offset by another stand-off, namely between the U.S. and China," PVM analyst Tamas Varga said in a note.
"The general consensus is that no break-through will take place towards the end of the week when the two leaders resume their trade talks. The likely failure to reach a mutually acceptable trade agreement will raise demand concerns that will dishearten oil bulls."
Demand concerns weighing on oil prices were briefly overcome last week when Brent climbed 5% and U.S. crude surged almost 10%, its strongest week since 2016, after Iran shot down a U.S. drone on Thursday, adding to tensions stoked by attacks on oil tankers in the area in May and June.
Washington has blamed the tanker attacks on Iran, which denies having any role.
U.S. President Donald Trump targeted Iranian Supreme Leader Ayatollah Ali Khamenei and other top Iranian officials with sanctions on Monday. Iran said this move closed the path of diplomacy.
Trump also said on Twitter that other countries should protect their own oil shipping in the Middle East rather than have the United States protect them.
Meanwhile, the Organization of the Petroleum Exporting Countries and its allies including Russia appear likely to extend a deal on curbing output when they meet on July 1-2.
The chief executive of Saudi Aramco, state oil firm of OPEC's de facto leader, said it had no plan to increase its current maximum output capacity of 12 million barrels per day (bpd), given sizeable spare capacity.
Russian Energy Minister Alexander Novak said international cooperation on crude production had helped stabilize oil markets and was more important than ever. He also voiced concerns about demand.
U.S. sanctions on Iran and Venezuela have cut oil exports from the two OPEC members but U.S. production has been rising, leading some Russian officials to accuse Washington of carving out market share for its energy exports.
(Additional reporting by Aaron Sheldrick in TOKYO, editing by Louise Heavens)