Reykjavik agreed in June to reimburse the two countries for compensating people holding so-called Icesave accounts at Landsbanki, one of three top Icelandic banks that failed last year under the weight of massive debts.
Some 300,000 people in Britain, and others in the Netherlands, had funds deposited in the accounts.
The bill now in parliament could leave Icelandic taxpayers footing much of the cost and is widely unpopular among politicians and the public, already anxious about the decision last week to seek to join the EU.
The decision by parliament effectively postpones a vote on the bill at least until the assembly reconvenes in early August.
A parliament official said the decision was made to give the assembly's budget committee more time to deliberate on the issue and allow staff some holiday after an exhausting session that has lasted far longer than planned.
Earlier on Thursday, Icelandic Foreign Minister Ossur Skarphedinsson told reporters in Stockholm he was confident enough support would be mustered for parliament to pass the bill.
Iceland last week applied for EU membership after a close vote in parliament. The dispute with Britain and the Netherlands is seen as one of hurdles to accession.
"I wouldn't say it will have an overwhelming support, but I think it [the bill] will be passed with enough votes for me to be happy," Skarphedinsson said, speaking after a ceremony to mark Iceland's EU application. Sweden currently holds the six-month EU presidency.
Restructuring the banking sector and repaying creditors are seen as key to reviving an economy in the clutches of a deep recession and placating the International Monetary Fund and others that have pledged $10 billion toward Iceland's recovery.
On joining the EU, Icelanders are torn between feeling it may be best for the country and fears they could lose their independence to faraway Brussels.
Joining would require Iceland to sign up to EU's centralised fisheries policy and cede control of waters teaming with cod, haddock and herring. Icelandic officials have said they could seek some flexibility since its territorial waters do not border those of any EU member state.
Recent opinion polls have indicated a small majority in Iceland in favour of joining the Union.
(EurActiv with Reuters.)
