In the letter, which is dated Monday and was sent to the European Central Bank, the European Commission and the International Monetary Fund, Schäuble calls for "a quantified and substantial contribution of bondholders to the support effort" for Greece.
"Such a result can best be reached through a bond swap leading to a prolongation of the outstanding Greek sovereign bonds by seven years," Schäuble wrote.
German authorities have insisted on private sector involvement in any future Greek rescue plans but had not given details on what they expected until now.
The finance minister also evoked "a real risk of default within the eurozone" if Greece is not granted the next slug of aid contained in an initial rescue plan worth €110 billion ($160 billion) backed by the European Union and International Monetary Fund.
He added that "I see a need to reach agreement on a new programme for Greece to ... prevent a bankruptcy. I am aware that discussions on private sector participation continue, but I am sure they will result in a constructive solution before our meeting on June 20," Schäuble said.
He referred to a gathering of eurozone finance ministers.
The ECB opposes forcing private investors to participate in a new Greek rescue plan, because it would be tantamount to restructuring the country's debt, which could have serious consequences for the eurozone as a whole.
But ECB officials have indicated they could live with a roll over of Greek debt on a voluntary basis, which would involve banks and other investors agreeing to buy new bonds to replace ones set to mature in the next couple of years.