By Josie Cox 

European stocks surged Wednesday on renewed optimism about a possible deal between Greece and its creditors.

The Stoxx Europe 600 rose sharply after Greek Prime Minister Alexis Tsipras sent a new proposal for budget cuts and policy overhauls to Greece's creditors as part of a request for a new bailout.

However, the proposal falls short of the demands of his country's creditors, European officials said Wednesday.

The pan-European index was up 2.1% midafternoon after two sessions of heavy losses. Germany's DAX climbed 2.7%. France's CAC added 2.5%.

Lyn Graham-Taylor, a rates strategist at Rabobank in London, said the latest Greece developments still leave a lot of questions unanswered but that investors are generally cheered by the fact that "negotiations seem to have resumed."

"It looks like things are coming to a head one way or another," he added, saying that this was encouraging some of those who had been waiting on the sidelines to tentatively re-enter the market.

In the U.S., the S&P 500 rose 0.8% in early trade.

Wednesday's rally extends a roller-coaster ride for European stocks over the past month. As hopes for a Greek deal have ebbed and flowed, markets have been hit by large daily swings.

A 2.3% surge for the Stoxx Europe 600 on June 22, the largest since May, was wiped out by the biggest one-day fall since October on Monday. Swings in other indexes, like Germany's DAX and Italy's FTSE MIB, have been even larger.

The moves come during a strong year for European equities. The Stoxx Europe 600 remains more than 13% higher in 2015, despite a 3% loss over the past month.

"The markets don't like uncertainty and some of that uncertainty is now starting to lift," said Philip Lawlor, a partner at London-based Smith & Williamson Investment. "The pressure is on and it's inevitable that something will have to give soon."

Some warned against gleaning too much from the latest twists and turns in Greece's debt crisis.

"The market is reading the latest headlines as signs that Tspiras is backing down, but we have to remember that he is only doing so on some points," said Nick Lawson, a managing director in Deutsche Bank's equity team.

He also said the market might be reacting so strongly because investors are "scared of missing the upside, especially considering the fall we have had already this week."

Data from stock-exchange operator BATS Chi-X Europe showed that trading volumes in European stocks were high Wednesday.

Mr. Tsipras is expected to make a statement on television later in the day. Eurozone finance ministers were to discuss the new Greek requests in a conference call in the late afternoon.

Bonds yields fell in Italy, Spain and Portugal, highly indebted countries that in past years were seen as vulnerable to spillover from Greece.

The yield on 10-year Italian debt was down 0.06 percentage point at 2.24%, while the yield on Spanish 10-year debt was down 0.04 percentage point at 2.22%. Portuguese 10-year debt was yielding 2.85%, 0.10 percentage point lower on the day. Yields rise as prices fall.

The yield on the German 10-year bond, considered a safer asset, was 0.06 percentage point higher on the day at 0.83%.

The euro was down 0.6% against the dollar at $1.1064.

On Sunday, the Greek public will decide whether to accept the terms that creditors are offering in return for more bailout funds.

Gary Jenkins, chief credit strategist at asset manager LNG Capital, said the latest proposal may be a sign that Mr. Tsipras is getting "cold feet" about the referendum.

"He may be realizing that if he loses the referendum, he loses everything."

Overnight, Greece became the first advanced economy to default on a payment to the International Monetary Fund. However, this was widely anticipated.

Greek stock markets will remain closed this week and on Tuesday bond-trading platform Tradeweb said it had blocked trading in a number of Greek government bonds after a notification from the U.K. regulator.

Eric Lascelles, chief economist at RBC Global Asset Management, said that Greece is still "slightly more likely to remain in the eurozone than exit," adding, however, that "the risks are growing."

Tommy Stubbington contributed to this article.

Corrections & Amplifications

The Stoxx Europe 600 surged 2.3% on June 22. An earlier version of this article incorrectly said the 2.3% surge was last Wednesday.

Write to Josie Cox at josie.cox@wsj.com