The 10 billion euro ($12.88 billion) EU bailout required that 5.8 billion euros ($7.47 billion) be seized from Cypriot bank accounts.
The Financial Times said a last-minute attempt to get funds from the Kremlin instead of the account seizures stunned EU leaders in Brussels. The seizures would have taxed deposits of more than 100,000 euros ($128,800) in Cypriot banks -- accounts mainly held by Russians.
"It became clear that there are these business interests in Parliament who are telling us that we need to protect the non-resident depositors," an unnamed eurozone official told The Financial Times.
Cyprus faces the prospect of a full banking meltdown and possible exit from the eurozone without a bailout, the newspaper said.
Earlier, Cyprus raced to rework the $13 billion bailout's terms to ease or scrap a tax on small bank depositors while seizing more from larger depositors, officials said.
Cypriot banks were closed Tuesday by government order and are to remain closed until Thursday in an unprecedented step to avoid a bank run when large numbers of customers withdraw their deposits at the same time.
Many Cypriots have been withdrawing money from ATMs since Saturday, when news broke of the surprise decision by wealthy European nations -- led by German Chancellor Angela Merkel, who faces re-election this year -- to force bank depositors to share in the Cyprus eurozone bailout cost.
The decision upended an EU practice to finance eurozone bailouts largely by European taxpayers.
EU leaders decreed when a bank or country goes broke, bond investors and even bank depositors will have to pay a meaningful part of the bill.
In a bid to calm fears among Cyprus' 800,000 people, the country's central bank Monday ordered banks to restock cash machines while both European and Cypriot central bank officials developed contingency measures to deal with large deposit outflows.
Thousands of Cypriots staged noisy demonstrations in Nicosia, including outside the Parliament building.
The 56-member Parliament put off until Tuesday afternoon a debate on the deposit tax, Parliament speaker Yiannakis Omirou said.
Parliament's failure to back the rescue deal, or a bank run in Cyprus, is widely expected to revive fears over the eurozone's integrity.
Cypriot President Nicos Anastasiades said on national TV Sunday if Parliament did not approve the bailout terms, it would mean a "complete collapse" of Cyprus' banking sector, major losses for depositors and businesses, and a possible exit of Cyprus from the eurozone.
Several revamps of the bailout terms were discussed Monday and Tuesday -- all seeking to find another way of raising 5.8 billion euros, or about $7.5 billion, needed for Cyprus' bailout so officials could lower or even eliminate the tax on smaller deposits.
European officials told The Wall Street Journal a new proposal would let depositors with 100,000 euros or less be taxed at a 3 percent rate -- down from 6.75 percent originally proposed.
A euro was worth a little less than $1.29 in Tuesday trading.
Depositors with 100,000 euros to 500,000 euros would be taxed at 10 percent, and those with more than 500,000 euros would be taxed at 15 percent, the newspaper said.
The original deal was for Cyprus to tax those with more than 100,000 euros at 9.9 percent.
The government was also considering a plan to eliminate the tax on depositors with less than 20,000 euros, the Journal said.
The Financial Times cited a senior EU official as saying international lenders were pushing Cyprus to exempt all deposit holders with less than 100,000 euros.
The so-called troika of the European Central Bank, European Commission and International Monetary Fund would compensate for that by taxing all deposits above that level at 15.6 percent, The Financial Times said.
Thousands of Cyprus' bigger depositors are Russian, with an estimated $19 billion in Cypriot banks.
Cyprus has styled itself as a tax haven to attract international deposits. Its banking system is now estimated to have deposits amounting to five to eight times the size of the island's $22 billion economy.
A spokesman for Russian President Vladimir Putin attacked the proposed tax-the-depositor plan as "unjust, unprofessional and dangerous."
Russia suggested a separate but crucial $3.3 billion loan to Cyprus could now be in doubt.