* Greece submits new aid proposal to creditors
* June U.S. consumer confidence rises more than expected
* Home prices rise slower than expected in April
* Indexes up: Dow 0.13 pct, S&P 0.27 pct, Nasdaq 0.57 pct (Adds comments, updates prices, trading volume)
By Sinead Carew
June 30 (Reuters) - U.S. stocks finished up after a choppy trading day as investors held out hope on Tuesday for deal to keep Greece in the euro even as it veered close to a potential debt default.
Greece, hours from missing a 1.6 billion euro ($1.8 billion) payment due to the International Monetary Fund, submitted a new 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.
The Greek government indicated that it could change its stance on a referendum it scheduled for Sunday if a new loan request could be agreed, euro zone sources told Reuters.
After starting with a rally and then turning negative along with European equities, Wall Street reversed course again in the afternoon on hopes that Greece's Prime Minister Alexis Tsipras was showing signs of willingness to talk.
"I think (Tsipras) was a little defiant and arrogant and he overplayed his hand," said Kenny Polcari, director of the NYSE floor division at O'Neil Securities in New York. "It could still blow up. There's still nervousness in the market."
U.S. corporations have limited exposure to Greece, but investors are concerned about the fallout across Europe if the country exits the euro zone.
Volatility picked up on Tuesday also due to the expiration of quarterly options. The expiry of weekly options is a day early due to U.S. Independence Day holiday observance on Friday.
On Monday, U.S. stocks had fallen sharply in heavy trading and the S&P 500 and the Dow had their worst day since Oct. 9 due to worries about Greece.
The Dow Jones Industrial Average index ended Tuesday showing a decline for the first half of 2015. The S&P 500 benchmark showed its first quarterly decline in ten quarters but Nasdaq showed its 10th consecutive quarter of gains.
The S&P, the Dow and Nasdaq all closed lower for the month.
Dramatic headlines out of Greece have overshadowed U.S. data so far this week. Single-family home prices rose in April from a year earlier but at a slower pace than forecast, a closely watched survey said on Tuesday.
A separate report showed the U.S. consumer confidence index rose more than expected to 101.4 in June. The consensus was for a reading of 97.3.
The Dow Jones industrial average rose 23.16 points, or 0.13 percent, to 17,619.51, the S&P 500 gained 5.48 points, or 0.27 percent, to 2,063.12 and the Nasdaq Composite added 28.40 points, or 0.57 percent, to 4,986.87.
Seven of the 10 major S&P 500 sectors rose, with the telecommunications services, utilities and consumer staples sectors showing very slight declines.
Advancing issues outnumbered declining ones on the NYSE by 1,894 to 1,226, for a 1.54-to-1 ratio on the upside; on the Nasdaq, 1,742 issues rose and 1,058 fell for a 1.65-to-1 ratio favoring advancers.
The benchmark S&P 500 index was posting 4 new 52-week highs and 25 new lows; the Nasdaq Composite was recording 30 new highs and 108 new lows.
About 7.5 billion shares changed hands on U.S. exchanges, compared with the 6.3 billion average for the last five sessions, according to data from BATS Global Markets. (Additional reporting by Siddharth Cavale in Bengaluru, Ryan Vlastelica and Caroline Valetkevitch in New York; Editing by Meredith Mazzilli)