Greece is fast-approaching bankruptcy and is due to make a payment of €1.5bn (£1.09bn) to the International Monetary Fund (IMF) on 5 June.
Mr Varoufakis told Star TV a deal with creditors was “very close” and denied the country might leave the eurozone.
“Another currency is not on our radar,” he added.
The Greek government, EU and IMF have been locked in negotiations over economic reforms they say must be implemented before the latest €7.2bn tranche of the country’s bailout fund is released.
Issues over pension reform, deregulation of the labour market, and the re-hiring of 4,000 former civil servants are yet to be resolved.
The deadlock has led to concerns that Greece could run out of cash.
Last week, the government raided its IMF reserves in order to pay €750m in debt interest on its existing loans.
Read More: BBC