"Greece has made enormous strides at reaching a deal… It is now up to institutions to do their bit. We have met them three-quarters of the way, they need to meet us one-quarter of the way," Varoufakis told BBC.
The finance minister also said that if Greece leaves the Eurozone as a result of the crisis it would be a "disaster for everyone involved."
According to the broadcaster, Greek Interior Minister Nikos Voutsis said that the country does not have money to repay some $1.7 billion to the International Monetary Fund (IMF) in June.
The rest of the June repayments are in short-term bills.
The IMF, the Eurozone and the European Central Bank are Greece's biggest lenders.
On Saturday, the country's Prime Minister Alexis Tsipras called on creditors to make concessions as Greece was ready to accept a viable agreement but not on "humiliating" terms.
In February, Greece pledged to reform its ailing economy as part of the deal with the Eurozone finance ministers, who agreed to extend the country's bailout for four months.
