"We need a solution as soon as possible so that we have clarity ... We have been talking about this for the whole of May and June, discussing the same issues again and again without resolving them," Merkel said.
"Germany and France are determined at the upcoming EU summit ... to say that we want a quick solution," Merkel told reporters in Berlin after talks with French President Nicolas Sarkozy in Berlin.
Sarkozy agreed, saying: "France and Germany want this new programme to be worked out as quickly as possible. There is no time to lose," he said. Both said that private investors should be involved in the new package on a voluntary basis.
"We want involvement of the private sector on a voluntary basis. I want to stress this. There is no legal basis so far for there being obligatory involvement," Merkel said.
"This must be worked out together with the ECB (European Central Bank) so that there are no contradictions with the ECB," she said.
Merkel heads a group of top eurozone economies that had been pressing for private investors to contribute up to a third of the second rescue package by accepting later repayment on their Greek bonds.
But Sarkozy backs the ECB and the European Commission, which want the private sector to contribute on a "voluntary" basis, hoping thereby to avoid any action that ratings agencies might deem a default and which would likely cause chaos throughout the eurozone.
Amid a raft of warnings that Berlin's terms for the eurozone's fourth emergency bailout could trigger a potentially disastrous Greek debt default, Merkel appeared to back down after the talks with Sarkozy in Berlin.
Yields on Greek bonds fell and the euro rose, with UniCredit economist Andreas Rees saying Merkel and Sarkozy had "made a first step towards compromise and to salvaging (Europe's) capacity to act."
"We are well aware of these worries, which is why we are calling for a solution as soon as possible," Merkel, the head of Europe's top economy and the biggest contributor to the eurozone's three bailouts so far, told reporters.
Merkel said she now backed a new package for Athens along the lines of a deal on Romanian debt agreed in Vienna in 2009, whereby private banks agreed to buy new government bonds to replace ones that matured.
"The Vienna Initiative, as it is known, is a good foundation and I believe we can achieve something on this basis," Merkel said.
This "rollover" option is favoured by the ECB and France, since it avoids the risk of rating agencies declaring Athens in default, something which could send shock waves through the European and global financial systems.