Intense negotiations are under way between political parties amid public anger at a one-off levy of up to 10% being imposed on savers.
President Nicos Anastasiades said refusing the bailout would have led to the collapse of the country's  banks.
The speaker of the European Parliament has called for the levy to be revised to protect small-scale bank customers.
The 10bn-euro ($13bn; Â£8.6bn) deal agreed by the EU and IMF in Brussels marks a radical departure from previous international aid packages.
Under its terms, people in Cyprus with less than 100,000 euros in their accounts would have to pay a one-time tax of 6.75%. Those with sums over that threshold would pay 9.9% in tax.
Depositors will be compensated with the equivalent amount in shares in their banks.
It is believed that eurozone leaders, particularly in Germany, insisted on the levy because of the large amount of Russian capital kept in Cypriot banks, amid fears of money-laundering.
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President Anastasiades, who was elected only last month, is due to address the country shortly (from 18:00 GMT).
He said in a statement earlier that Cyprus had had to choose between the "catastrophic scenario of disorderly bankruptcy or the scenario of a painful but controlled management of the crisis".
The president has been meeting with members of the parliament's finance committee, his office said.
On Saturday the head of the committee, Nicholas Papadopoulos, expressed shock at the deal, saying it was "much worse than expected".
The president's Democratic Rally party - which has 20 seats in the 56-member assembly - needs support from other factions to ratify the bailout.
A spokesman for one of its coalition partners, the Democratic Party, told BBC News they wanted assurances that the deal would resolve the problems facing Cyprus before voting in favour.
Opposition leader George Lillikas, an independent, said the president had "betrayed the people's vote".
The speaker of the European Parliament, Martin Schulz, argued in a newspaper interview that there should be an exemption from the levy for savers, for example, who had less than 25,000 euros in their accounts.
By Gavin HewittEurope editor
"The solution must be socially acceptable," Mr Schulz, who belongs to Germany's opposition Social Democrats,  told Germany's Welt am Sonntag newspaper.
The German Chancellor, Angela Merkel, defended the levy.
"With this deal, the responsible people are partly included in those countries and not only the taxpayers in other countries," she said at a party meeting in her home constituency.
"And I think it's right that we went down that road and I think it's a good step which will certainly make it easier for us to approve the help for Cyprus."
As with past eurozone bailouts, the deal must be approved by the lower house of parliament in Germany, the EU's biggest economy.
If the levy goes ahead, it will affect many non-Cypriots with bank accounts, including UK expatriates.
However, depositors in the overseas arms of Cypriot banks will not be hit. Bank of Cyprus UK and Laiki Bank UK both confirmed on their websites that there would be no impact.
Chancellor George Osborne said the UK would compensate any government employees and military personnel whose bank accounts were affected.
The levy itself will not take effect until Tuesday, following a public holiday, but action is being taken to control electronic money transfers over the weekend.
Cyprus's parliament has postponed an emergency session on a controversial bailout deal for the country's banks.