* Euro zone stocks fall, Wall St holds to small gains
* Treasury yields edge up on hopes for a Greek deal
* China shares break losing streak (Updates with U.S. market open, adds comment, changes dateline from previous LONDON)
By Rodrigo Campos
NEW YORK, June 30 (Reuters) - Stocks on Wall Street edged up on Tuesday to keep a global equities gauge higher despite declines in Europe, while the euro remained under pressure as Greece grappled with a looming deadline on a debt repayment to the IMF.
Greece submitted a new two-year aid proposal to its creditors, calling for debt restructuring in what seemed like a last-ditch effort by Athens to resolve its impasse with lenders. Talks between Greece and its creditors broke down over the weekend and forced Greece to close its banks and impose capital controls.
Greeks are due to vote in a referendum on Sunday that EU partners say will amount to a choice between staying in the euro or leaving.
Stocks tumbled in major markets on Monday after the Greek talks broke and Tuesday's fading advance was seen as a sign of lingering hope of a deal that would help avoid a default.
"Even after these market swings, a Greek exit is still not fully discounted as a positive outcome is still possible," BNP Paribas Investment Partners said in a note to clients.
"With a majority of Greeks in favor of staying in the euro zone, there is a decent probability of a referendum outcome in favor of the creditors' proposals. But until the results are known, we are likely to see continued market volatility."
The Dow Jones industrial average rose 32.12 points, or 0.18 percent, to 17,628.47, the S&P 500 gained 3.61 points, or 0.18 percent, to 2,061.25 and the Nasdaq Composite added 13.83 points, or 0.28 percent, to 4,972.29.
The S&P earlier rose as much as 0.8 percent.
Equities gained also from support out of China, after signs of intensifying government support helped stop a sharp slide. The Shanghai Composite Index rallied 5.6 percent while Hong Kong's Hang Seng added 1.1 percent.
The MSCI All-Country World equity index was up 0.3 percent. The pan-European FTSEurofirst 300 index and the euro zone's blue-chip Euro STOXX 50 index were down 0.8 percent and 0.5 percent respectively.
Despite the large moves out of risk assets on Monday and the massive volumes, panic was considered to be avoided, with market participants citing Europe's improved ability to fight financial contagion.
However, the borrowing cost on a key source of overnight loans for Wall Street jumped as traders competed for a shrinking pool of cash before quarter end. The interest rate on overnight loans in the repurchase agreement market was last quoted as high as 0.65 percent, which would be the highest closing level since November 2008.
In commodities, oil futures bounced back from three week lows but Brent was set to close its second consecutive month of losses.
"Markets are worried that a Greek debt default could hit European economic growth and thus fuel demand," said Tamas Varga, oil analyst at London brokerage PVM Oil Associates.
"Capital controls are not bullish for the economy, and not bullish for oil. How much can you spend on gasoline if you can't withdraw more than 60 euros a day?"
Brent was up 1.9 percent at $63.20 a barrel after falling to $61.35 on Monday, its weakest since June 5. It was down 3.7 percent for June. U.S. crude gained 1 percent at $58.91. It was set for its first monthly decline in three, down 2.3 percent.
The euro was last down 0.44 percent against the U.S. dollar at $1.1185. Traders who had bet against the single currency continued to repurchase it after Monday's rally took it from a four-week low of $1.09550 to a nearly one-week high of $1.12790.
The euro's weakness is "partially the reality that there is not going to be a payment to the IMF," said Jason Leinwand, managing director at rates, currencies, and commodities derivatives hedge advisory firm Riverside Risk Advisors in New York.
He said, however, that optimism for a Greek resolution lingered given the upcoming referendum, limiting the euro's losses.
That same feeling permeated U.S. debt markets, and Treasuries yields rose slightly. Benchmark 10-year Treasuries notes were down 6/32 in price to yield 2.3531 percent, up 2 basis points from late on Monday.
The 30-year bond was down 11/32 in price for a yield of 3.1162 percent. (Reporting by Rodrigo Campos, Richard Leong and Sam Forgione in New York, additional reporting by Lionel Laurent and Christopher Johnson in London; Editing by Meredith Mazzilli)