Cypriots say they have been betrayed by Europe
Finance ministers from the eurozone have asked Cyprus to reduce the burden on small investors from a proposed levy on savings, linked to a bailout.
Plans for a one-off tax of 6.75% on savings up to 100,000 euros (£86,000; $130,000) have outraged Cypriots.
Banks in Cyprus are to remain closed until Thursday, as efforts to revise an international bailout package continue.
A parliamentary vote on the package has been repeatedly postponed, but is now expected on Tuesday.
The 10bn-euro bailout agreed with the EU and IMF demands that all bank customers pay a one-off levy.
The government's efforts to shift more of the burden onto wealthier depositors enraged Russians, who form the bulk of overseas investors and have deposits worth billions of dollars in Cypriot banks.
Russian President Vladimir Putin called the proposed levy "unfair, unprofessional and dangerous", and Moscow has expressed frustration Russia was not included in European decision-making on Cyprus.
Threat to confidenceUnder the currently agreed terms of the levy, depositors with less than 100,000 euros in Cyprus accounts would pay a one-off tax of 6.75%, while those with sums over that threshold would pay 9.9%.
But the move has outraged Cypriots and sparked heavy cash withdrawals from banks.
Since the start of the financial crisis there has been a guarantee that deposits under 100,000 euros in banks in the EU would be protected.
Many observers believe the Cypriot levy breaks the spirit of that agreement, and there is concern that it could also damage the confidence of depositors in other eurozone countries, reports the BBC's Chris Morris in Brussels.
Analysis
Cypriots will tell you they're a resilient nation. They bounced back from the war of 1974 and became a prosperous EU member three decades later.
But even they are feeling defeated by this shock tax. "Daylight robbery" is what many here call it.
"If Brussels insists on this, we should leave the EU altogether," one elderly gentleman told me in a Nicosia cafe.
And that is perhaps the lasting damage of this affair - a tiny yet proud EU member now feels bullied and blackmailed by the powerful, the old north-south division of Europe widening again.
Yet many argue Cyprus sleepwalked into this mess. For years it thrived as a tax haven, its banking sector eight times the size of its economy. The problem signs were there but few were willing to heed them.
Eurozone finance ministers - the Eurogroup - discussed the situation in a conference call on Monday evening.
Following the talks, its president Jeroen Dijsselbloem issued a statement saying the group "continues to be of the view that small depositors should be treated differently from large depositors and reaffirms the importance of fully guaranteeing deposits below 100,000 euros".
He said Cyprus would "introduce more progressivity in the one-off levy" - in other words, shift the burden away from small savers towards bigger depositors - provided that the same amount of funds, 5.8bn euros, was raised.
Mr Dijsselbloem urged "a swift decision by the Cypriot authorities and parliament to rapidly implement the agreed measures".
Vote 'close'President Anastasiades has been holding talks with ministers and MPs at the parliament building in Nicosia, where hundreds of people noisily protested on Monday.
The BBC's Mark Lowen in Nicosia says there are suggestions Mr Anastasiades may want to lower the former rate to 3%, while raising the levy on the larger depositors to 12.5%.
The debate and vote in Cyprus' parliament is now scheduled for 18:00 local time (16:00 GMT) on Tuesday. It was to have been held on Sunday.
- Depositors with under 100,000 euros deposited must pay 6.75%
- Those with more than 100,000 in their accounts must pay 9.9%
- Depositors will be compensated with the equivalent amount in shares in their banks
- The levy is a one-off measure
The president's Democratic Rally has 20 seats in the 56-member assembly and needs other parties' support to ratify the deal.
The vote remains too close to call, correspondents say.
Speaker Yiannakis Omirou, of the EDEK party, said: "Parliament is called to legalise a decision to rob depositors blind, against every written and unwritten law. We refuse to subscribe to this."
Mr Anastasiades insists that without the bailout Cyprus could face bankruptcy and a possible exit from the eurozone - a fear echoed by European officials.
The US has called for a "responsible and fair" resolution.
Protesters in Cyprus have held up banners blaming Germany for the controversial bailout deal, but Germany says it always favoured protecting bank accounts with up to 100,000 euros, and insists it was the Cypriot government, European Commission and ECB that decided on the levy terms.
Earlier European Commission spokesman Simon O'Connor defended the group's actions, saying its original decision on the bailout was "taken by unanimity, all the member states of the eurozone, including Cyprus".
Stock markets in the US, Asia and Europe fell in early trading, though some of their losses were recouped later in the day. The euro also fell.
Cyprus may only be a tiny fraction of the eurozone economy, our Brussels correspondent says. But the sense of uncertainty surrounding it is sending shivers through the financial markets.
