In an interview with German Sunday newspaper Frankfurter Allgemeine Sonntagszeitung he said that "Greek people and their elected lawmakers" will have to decide if they want to abandon the euro currency.
Pending those political decisions, central banks "must ensure that the risk on our balance sheets remains manageable," Weidmann was quoted as saying.
If Greece were to abandon the euro, the European Central Bank, national central banks and taxpayers would face high losses that analysts say could add up to some €200 billion ($255 billion).
"Indeed, I would not consider it to be right for the euro system to further increase its level of risk for Greece," Weidmann said.
Greece is set to hold elections on June 17 to end a political deadlock after a previous vote on May 6 produced a hung Parliament, with the country's future in the 17-nation eurozone potentially at stake.
Discussing the prospect of Athens leaving the eurozone had long been a taboo for European policy makers, but the issue rose to the forefront of policy debates after parties rejecting the international bailout agreements scored high results in this month's Greek legislative elections.
Edgy markets and analysts fear a victory next month by parties opposed to those austerity and reform commitments could spur bailout creditors to stop the flow of rescue loans, which would eventually force Greece to abandon the currency union, plunging the country into yet deeper recession and sending shockwaves through the global financial system.
Weidmann, who is also an influential member of the ECB's rate-setting council, warned Greece would suffer the most if the nation were to reject the bailout agreements and leave the euro.
"Such a decision would certainly hit Greece the hardest," he was quoted as saying.
Weidmann reiterated that Greece cannot expect further assistance from its partners if it fails to live up to the bailout agreements struck between previous governments and the so-called troika of creditors — the ECB, the EU Commission and the International Monetary Fund.
Greece has been surviving since May 2010 on an initial €110 billion package of rescue loans, and it was granted a second bailout of about €130 billion this year to keep the country afloat for the coming years.
"It's of fundamental importance that there can be no help without Greece itself making the agreed-upon efforts," Weidmann was quoted as saying, adding that this was also important in light of similar agreements with other countries and the stability of the euro as a whole.
Juergen Baetz can be followed on Twitter at www.twitter.com/jbaetz