To satisfy its creditors, the country needs to produce €1.6 billion by the end of Tuesday.
It's a big ask.
The world wide web has thrown up all sorts of unconventional strategies for keeping the country afloat, including the idea that Apple step in and buy it.
Apple's €217 billion cash hoard could put a pretty large dent in Greece's debt, but the tech giant is not in the market for a country, said CEO Tim Cook.
Step in plucky Briton Thom Feeney, who set up a crowdfunding page asking all of Europe to have a look down the back of the sofa.
"The European Union is home to 503 million people, if we all just chip in a few euro then we can get Greece sorted and hopefully get them back on track soon. Easy."
Easy? Well, in theory, if everyone in the EU, minus the population of Greece, donated €3.25, it would cover Tuesday's payment.
To pay off Greece's debt in full? Everyone in the EU would need to donate €550.
There are incentives though. A €3 donation will get you a postcard of Alexis Tsipras, apparently.
At the other end of the scale, a €1 million donation wins you the eternal gratitude of the Greek people.
(It was not certain whether Mr Feeny was in a position to deliver these prizes.)
It's a nice thought, but unlikely to be a magic debt bullet. At the time of writing, €18,577 has been donated in a day, by 1,326 charity-minded people.
At that rate, it would take another 236 years to meet Tuesday's repayment, and 40,000 years to pay off Greece's overall debt.
"It might be a short while," Mr Feeney admits.
Months of negotiations have failed to produce an agreement on Greece's massive €271 billion (£193bn; $303bn) debt.