In midday trade, Frankfurt's DAX 30 climbed two percent to 11,166.67 points and the CAC 40 in Paris advanced by 2.39 percent in value to 4,904.90.
Madrid won 1.63 percent and Milan gained 2.41 percent, while the Athens stock markets remained shut.
Outside the eurozone, London's benchmark FTSE 100 index rose 1.39 percent to 6,611.31 points compared with Tuesday's close.
In foreign exchange, the euro dipped to $1.1125 from $1.1139 late in New York on Tuesday.
European indices had slid on Tuesday, extending heavy losses seen at the start of the week, as hopes faded of a last-minute deal to stave off Greece defaulting on its payment to the International Monetary Fund.
The Greek government confirmed Wednesday it had sent an "amended" proposal to its international creditors in the hope of sealing a deal to stave off financial ruin.
A letter sent by Prime Minister Alexis Tsipras to the EU, ECB and IMF Tuesday evening asked for "a new agreement that regulates the country's financing issues through the European Stability Mechanism (ESM), to ensure debt sustainability by focusing on growth," a government source said.
Banks' share prices were rising strongly, with Royal Bank of Scotland up 3.30 percent, Commerzbank winning 2.66 percent and BNP Paribas increasing its value by 2.95 percent from Tuesday's close.
The European Union will meanwhile decide Wednesday whether to hand Greece a last-minute bailout package to prevent the country moving further towards an exit from the eurozone.
Greece on Tuesday failed to make a 1.5 billion euro ($1.7 billion) payment to the International Monetary Fund, becoming the first industrialised country to do so.
On the same day, the international bailout keeping the Greek economy afloat formally expired.
"Investors are still hopeful of a positive outcome," said Mike van Dulken, head of research at Accendo Markets.
"Suggestions are that fresh Greek proposals are now closer to those of the creditors."
Wednesday meanwhile marked the start of the third quarter after a volatile six months' trading for Europe's stock markets.
"It's not been a good quarter for European markets, posting fairly hefty falls, after a spectacular start to the year but they still remain (mostly) in positive territory for the year," said Michael Hewson, chief market analyst at CMC Markets UK.
Hewson noted however that London's FTSE 100, consisting of major energy and mining companies, had been additionally weighed down by lower commodity prices this year.
Asian stock markets mostly closed higher Wednesday, with Tokyo lifted by an upbeat survey on business confidence, but Shanghai's volatile run continued, plunging more than five percent.
US stocks finished moderately higher Tuesday, recovering some of the prior session's losses.