Eurozone finance ministers, known collectively as the Eurogroup, are confident of agreeing a bailout for Cyprus by the end of March.
The bailout could be worth up to 17bn euros ($22bn; £15bn).
Eurogroup head Jeroen Dijsselbloem said ministers were ready to help Cyprus, but the details still needed to be worked out.
Cyprus's new government has agreed to a review of how banks are implementing anti-money laundering laws, he said.
This is likely to appease Germany, which had raised concerns about money-laundering on the island.
"We agreed to target a political endorsement of the programme towards the second half of March," Mr Dijsselbloem said, referring to the rescue package.
EU Commissioner for Economic Affairs Olli Rehn hailed the breakthrough on money-laundering.
He had warned over the weekend that Cyprus leaving the eurozone remained a dangerous possibility.
"Even if you come from a big EU country, you should be aware that every member of the eurozone is systemically relevant," Mr Rehn was quoted as saying in Germany's Der Spiegel magazine.
"If Cyprus becomes disorderly insolvent, it is very likely that would lead to it exiting the eurozone."
Cyprus's newly elected President, Nicos Anastasiades, has promised to secure a financial bailout, saying his top priority is to restore the country's credibility.
Cyprus requested a bailout in June last year but the previous communist-led government was unable to reach an agreement.
Finance ministers from the 17 eurozone members held talks for about four hours on Monday, which largely focused on how to fund a potential rescue for the island.
In a statement, Eurogroup ministers welcomed "the commitment of President Anastasiades, reiterated by [Finance] Minister Sarris, to closely co-operate with Cyprus's European partners towards the earliest possible completion of the loan agreement."
Germany has pushed for depositors in Cypriot banks to help pay for the rescue, a process known as a "bail-in".
But Cyprus fears a b