Eurozone finance ministers have rejected a Greek government call to extend its bailout, just hours before it expires and a €1.6bn (£1.1bn) payment to the IMF falls due.
The move came after ministers held an emergency conference call.
Greek PM Alexis Tsipras had requested a short extension to Greece's current bailout, and a two-year rescue deal.
If it fails to repay the International Monetary Fund (IMF), Greece could risk leaving the euro.
Greece has also asked the IMF for more time, according to Deputy Prime Minister Yannis Dragasakis.
Eurogroup chairman and Dutch Finance Minister Jeroen Dijsselbloem said it would be "crazy" to extend the Greek bailout beyond its midnight expiration as Athens was refusing accept the European proposals on the table.
Speaking after the conference call, he added that a Greek request for a new €29.1bn European aid programme would be considered later.
The European Commission - one of Greece's "troika" of creditors along with the IMF and the eurozone's European Central Bank - wants Athens to raise taxes and cut welfare spending to meet its debt obligations.
Amid fears of a Greek default on its huge public debt of €323bn, people have queued at cash machines. Withdrawals are capped at just €60 a day.
Greek banks did not open this week after talks between Greece and its creditors broke down. However, up to 1,000 bank branches will re-open from Wednesday to allow pensioners - many of whom do not use bank cards - to withdraw up to €120.
Analysis: By Andrew Walker, economics correspondent
The sky will not fall in when the deadline for Greece to repay to the IMF passes, but the tension will rise.
If a payment to the IMF is missed and managing director Christine Lagarde informs her board, the eurozone would have the right under the loan agreements to demand immediate repayment of more than €180bn they have already lent Greece, together with interest.
Failure to repay the IMF comes under a list of "events of default" set out in the legal documents. IMF officials say Ms Lagarde intends to inform the board promptly if Greece doesn't pay on time.
Of course, in practice, the rest of the eurozone won't demand the money back and send in the bailiffs. Athens couldn't possibly repay and in any event they want to keep talking, hoping they can keep Greece in the eurozone. But the fact that they would have the option is a sign of how close Greece is getting to the brink.
On Tuesday evening, thousands of pro-EU protesters braved stormy weather and gathered outside the Greek parliament in Athens to urge a "yes" vote in a referendum on Sunday over whether the country should accept its creditors' proposals.
It follows a similar demonstration by those advocating a "no" vote - the path preferred by Mr Tsipras - on Monday.
EU leaders have warned that a no vote rejecting creditors' proposals would mean Greece leaving the eurozone - though Mr Tsipras says he does not want this to happen.
The ECB is believed to have disbursed virtually all of its emergency funds for Greece, amounting to €89bn (£63bn).
What will happen next?
01:00 Greek time Wednesday (22:00 GMT): Greece's €1.6bn repayment to the IMF is due.
5 July - the referendum on creditors' proposals, and many say Greece's membership of the eurozone, takes place
20 July - Greece must redeem €3.46bn of bonds held by the European Central Bank. If it fails to do so, the ECB can cut off Greece's access to emergency loans.
Days of turmoil
- Friday evening: Greek prime minister calls referendum on terms of new bailout deal, asks for extension of existing bailout
- Saturday afternoon: Eurozone finance ministers refuse to extend existing bailout beyond Tuesday
- Saturday evening: Greek parliament backs referendum on creditors' proposals for 5 July
- Sunday afternoon: ECB says it is not increasing emergency assistance to Greece
- Sunday evening: Greek government says banks to be closed for the week and cash withdrawals restricted to €60
- Monday evening: Greek prime minister appeals to Greeks to reject the creditors' proposals, saying this will give Greece "more powerful weapons" in negotiations; he also hints he will resign if the result of the referendum is a "yes" vote
Lenders' proposals - key sticking points
- VAT (sales tax): A new system to come in from 1 July, with three rates, aimed at boosting annual revenue by 1% of total output (GDP)
- Most goods to be taxed at top rate of 23%, including restaurants and catering and processed foods
- Reduced rate of 13% for basic food, electricity, hotels and water
- Super-reduced rate of 6% for medicines, books and theatre
- End exemptions and eliminate VAT discounts for Greek islands
- Create strong disincentives to early retirement
- Move retirement age up to 67 by 2022
- End Ekas "solidarity" top-up grant that some 200,000 poorer pensioners get - immediate Ekas cut for wealthiest 20% of recipients, and cut completely by 2020
- Pensioners' healthcare contributions to rise to 6%, from 4%