(Corrects date in paragraph 2 to April 15 from June 5)

* Brent on track for second month of decline

* U.S. crude heading for first monthly decline in three

By Aaron Sheldrick

TOKYO, June 30 (Reuters) - Oil futures hovered below three-week lows on Tuesday as investors waited for developments in Greece following a bank shutdown, keeping them away from riskier assets and putting Brent crude on course for a second month of declines.

Brent crude futures LCOc1 were down 3 cents at $61.98 a barrel at 0624 GMT, after falling to $62.01 on Monday, their weakest finish since April 15. The contract is heading for its second straight monthly decline and is down almost 5.5 percent in June.

U.S. crude CLc1 dropped 14 cents to $58.19, having closed down $1.30 at $58.33 a barrel, its lowest settlement since June 8. It is set for its first monthly decline in three and has fallen about 3.5 percent this month.

"Greece is still the word," said Ben Le Brun, market analyst at OptionsXpress in Sydney. "That story doesn't look like stopping anytime soon."

Tens of thousands of Greeks hit the streets on Monday after waking up to shuttered banks, closed cash machines and a climate of rumours and conspiracy theories following the breakdown in talks between Athens and its creditors. ID:nL5N0ZF11E

Any resolution to the crisis is unlikely before a referendum on Greece's bailout is held on Sunday, after Prime Minister Alexis Tsipras announced the vote, wrong-footing European leaders and policy makers.

Investors are also looking at the U.S. government's June payrolls report on Thursday and talks on Iran's disputed nuclear programme going on in Vienna, Le Brun said.

The former may reinforce ideas that the U.S. Federal Reserve could raise interest rates as early as September, the first such hike in about 10 years.

The Vienna talks would continue past Tuesday's deadline for a comprehensive agreement intended to open the door to ending sanctions in exchange for limits on Iran's most sensitive nuclear activities for at least a decade, a senior U.S. official said on Tuesday. ID:nL5N0ZF4AR

Asian imports of Iranian crude rose to the highest level this year in May at 1.2 million barrels per day, according to final figures on Tuesday. But they held at slightly above 1 million bpd for the first five months of the year, the level allowed under sanctions on Tehran. ID:nL4N0ZF2O5

Still, Greece is the overriding concern for investors in most commodities markets, Le Brun said.

"It will probably be very headline driven in the short term," he said.

(Editing by Joseph Radford and Biju Dwarakanath) ((aaron.sheldrick@thomsonreuters.com; 81-3-6441-1320; Reuters Messaging: aaron.sheldrick.thomsonreuters.com@reuters.net))

Keywords: MARKETS OIL/