President Nicos Anastasiades has been meeting MPs to discuss the terms, with a crucial debate and vote in parliament postponed until Tuesday.
Cyprus banks are closed and will remain closed until Thursday amid the crisis.
The 10bn-euro ($13bn; £8.6bn) bailout agreed with the EU and IMF demands that all bank customers pay a one-off levy.
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.
Russian President Vladimir Putin called the proposed levy “unfair, unprofessional and dangerous”, his spokesman said. Russian banks and businesses have large deposits in Cyprus.
The US has called for a “responsible and fair” resolution.
But at a news conference, the leaders of Germany, France and the European Commission refused to make any detailed comment on the crisis in Cyprus, reports said.
The debate and vote in Cyprus’ parliament has now been postponed until 18:00 local time (16:00 GMT) on Tuesday. It was to have been held on Sunday.
The president’s Democratic Rally has 20 seats in the 56-member assembly and needs other parties’ support to ratify the deal.
Eurozone finance ministers – the Eurogroup – discussed the situation in Cyprus in a conference call on Monday evening.
Following the talks, its president Jeroen Dijsselbloem issued a statement saying the group backed fully guaranteeing deposits below 100,000 euros and said Cyprus would “introduce more progressivity in the one-off levy”, provided that the same amount of funds were raised.
He urged “a swift decision by the Cypriot authorities and parliament to rapidly implement the agreed measures”.
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”.
Key officials in the ECB, Germany and France have all insisted Cyprus is free to decide how to raise the 5.8bn euros, but that the money must be forthcoming.
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.
President Anastasiades has been holding talks with ministers and lawmakers at the parliament building in Nicosia, where hundreds of people protested on Monday.
The BBC’s Mark Lowen, in Nicosia, says there are suggestions Mr Anastasiades is looking at lowering the cost to those with smaller savings.
Under the currently agreed terms, depositors with less than 100,000 euros in Cyprus accounts would have to pay a one-time tax of 6.75%. Those with sums over that threshold would pay 9.9%.
Our correspondent says the president may want to lower the former rate to 3%, while raising the levy on the larger depositors to 12.5%.
An EU source told Agence France-Presse there could be a three-way split on the level of levy, grouped into accounts holding less than 100,000 euros, between 100,000 and 500,000 and more than 500,000.
Mr Anastasiades insists that without the bailout Cyprus could face bankruptcy and a possible exit from the eurozone.
But Finance Minister Anton Siluanov has expressed frustration that Russia was not included in European decision-making on Cyprus.
“We had an agreement with colleagues from the eurozone that we’d co-ordinate our actions [on Cyprus],” Mr Siluanov told Reuters news agency.
“So, we will consider the issue of restructuring of the loan taking into account our [future] participation in the co-ordinated actions with the European Union to help Cyprus.”
Russia may now think twice about extending the loan, says the BBC’s Steve Rosenberg in Moscow – and perhaps even more worrying for Nicosia, he says, some of those rich Russian depositors may now move their money elsewhere.
The Moody’s ratings agency estimates that, at the end of 2012, Russian banks had placed $12bn in Cypriot banks, with corporate deposits at $19bn.
That means Russian corporate and individual investors could lose up to $2bn.
Russia has strongly criticised the Cyprus bailout deal.
Presidential spokesman Dmitry Peskov said on Monday: “Assessing the possible decision of imposing additional tax by Cyprus on deposits, [President] Putin said that this decision, if taken, would be unfair, unprofessional and dangerous.”
Prime Minister Dmitry Medvedev said: “It looks simply like the confiscation of other people’s money.”
The Russian government gave Cyprus a 2.5bn euro loan in 2011, and Russia had indicated it would consider extending the period of its loan to Cyprus and restructuring repayments.