Crude oil prices rebounded on the last full trading day in June amid tempered optimism about the prospects for Greek and Iranian negotiations.

Brent crude oil prices started the trading day Tuesday up about 1.1 percent from the previous session's close to $62.72 per barrel. For the month, Brent crude oil is down 3.6 percent. West Texas Intermediate, the U.S. benchmark, gained 1 percent to $58.92 per barrel, down 1.8 percent from June 1.

Greece by the end of the day is to pay the International Monetary Fund around $1.8 billion or default. The Bank of Greece said Monday it would only honor transactions from the government, legal persons and social security funds through July 6.

The Greek government called for a referendum on offers extended by its creditors during the weekend. German Chancellor Angela Merkel said the door to Greece remains open, adding the entire European economy is in a "better position to master the crisis."

A depressed Greek economy cast a dark cloud over the recent global financial crisis, from which the world economy is still struggling to escape. If Greece defaults and the euro fails, Merkel said, "Europe fails."

Eurostat, the statistical office of the European Union, said regional annual inflation for June is expected to be 0.2 percent, down from 0.3 percent in May.

On Iran, U.S. Secretary of State John Kerry said from the negotiation table in Vienna that Iranian nuclear negotiations were moving forward, though talks may go beyond Tuesday's deadline.

More Iranian crude oil could enter a market already coping with too much supply should sanctions pressure drop in response to an eventual deal. Kerry's Iranian counterpart, Javad Zarif, said he was confident about the outcome of the talks in Vienna.

"I am here to get a final deal, and I think we can," he said.