By Tommy Stubbington and Josie Cox

Markets rallied Friday after new Greek proposals for policy overhauls and budget cuts appeared to be closer to creditors' demands, fueling investor hopes that the country could strike a deal this weekend to keep it in the eurozone.

The Stoxx Europe 600 ended the session 2.0% higher, building on Thursday's gains. The pan-European index has now clawed back all of the losses made after Sunday's referendum, in which Greek voters rejected a set of the creditors' austerity demands.

In the U.S., the S&P 500 was 1.2% higher shortly after the European stock market closed.

"Markets are celebrating the likelihood of an agreement this weekend," said Steven Englander, a senior currency strategist at Citibank.

Greek stock markets have been closed all week but a small exchange-traded fund that trades on the NYSE Arca and tracks Greek stocks, the Global X FTSE Greece 20 ETF, was up more than 7% Friday.

"Provided that Greece's creditors are convinced that [Prime Minister Alexis] Tsipras is serious about reforms, then a deal will be done by Sunday," said Demetrios Efstathiou, a strategist at ICBC Standard Bank.

Some leaders from the currency bloc voiced optimism about the latest proposals on Friday. Eurozone finance ministers and European leaders are set to assess the proposals during crisis meetings on Saturday and Sunday.

"A deal on Greece is likely this weekend," said Alberto Gallo, head of macro credit research at Royal Bank of Scotland. He added that nothing has been decided yet, but that the latest developments are very positive.

Germany's DAX added 2.9% and France's CAC-40 rose 3.1% on Friday. In southern Europe, Italy's FTSE MIB climbed 3.0% on the day while Spain's IBEX was 3.1% higher.

The relief was also felt in debt markets.

Bonds rallied in Spain, Italy and Portugal. Those countries have been rattled by the spillover from Greece's debt crisis, but 10-year yields in all three fell to their lowest levels in two weeks during the day.

Greek two-year bond yields tumbled from above 50% to just under 30%, according to Tradeweb, although trading in Greek government bonds remains limited. Yields fall as prices rise.

German debt, seen as a haven by investors, fell. German 10-year yields rose 0.16 percentage point to 0.89%.

The euro climbed 0.8% against the dollar to $1.115. Earlier in the session it hit its highest level against the buck so far in July and hit a one-week high against Japan's yen, which is also considered a safer asset during times of market instability.

Hurdles remain. Greece's plan may face tough domestic opposition. And the shutdown of Greece's financial system last week and uncertainty over its future in the eurozone have delivered fresh blows to the economy that would have to be offset by new cuts or extra revenue.

"I would rather wait before celebrating," said Luca Paolini, chief strategist at Pictet Asset Management, which has EUR149 billion ($164.7 billion) in assets under management. "But it now looks more likely than not Greece will stay in the euro."

If Athens can secure a new bailout, investors are likely to drive European stocks higher as they refocus on improving corporate earnings in the region, Mr. Paolini said.

Giovanni Zanni, an economist at Credit Suisse, said the situation was still very fluid and that even if there is a deal this weekend, "questions on implementation and ownership of reforms will continue."

In commodities markets, Brent crude oil was 0.1% lower at $58.53 a barrel. Gold was largely unchanged at $1,159.90 a troy ounce.

Write to Tommy Stubbington at tommy.stubbington@wsj.com and Josie Cox at josie.cox@wsj.com