Spain formally asked for $77 billion from its eurozone brethren. Cyprus did not specify the size of the bailout it was looking for, The New York Times reported.
Spanish Economy Minister Luis de Guindos made the request in a letter to Eurogroup President Jean-Claude Juncker, CNN reported.
In the letter, Guindos thanked the Eurogroup, a group of 17 countries that use the euro, for offering to support Spain on June 9 with as much as $125 billion.
"I have the honor to write to you, in the name of the Spanish government, to formally request financial assistance for the recapitalization of the Spanish banks that require it," Guindos wrote.
By using European aid funds, rather than through governments, Spain seeks to break a current link between bank risks and sovereign debt that helped push Spain's borrowing costs to record highs last week, advocates say.
The International Monetary Fund, a European rescue fund contributor providing technical assistance for the Spanish bank support, Thursday came out strongly in favor of letting rescue funds go directly to banks.
As a bailout condition, Spain's eurozone partners may require Spain's domestic banking sector to create one or more "bad banks" to house property assets and the forced liquidation of insolvent institutions, the Financial Times reported Monday.
Officials will consider basing this on restructuring examples from Ireland, where a central "bad bank" was created, and Germany, where "toxic" assets were placed in separate financial corporations alongside individual banks, the newspaper said.
Under current rules, eurozone rescue funds can't lend directly to banks, but some politicians say changing the rules may be relatively easy.
For Cyprus, the request puts a stain on what was to be a proud step forward into international affairs, as the country takes its turn in a few days to take over the European Union presidency, which is done on a rotating basis.
"The purpose of the required assistance is to contain the risks to the Cypriot economy, notably those arising form the negative spillover effects through its financial sector," the government said in a statement.
In Greece, unexpected health problems sidelining the country's new prime minister and finance minister put off a planned Monday visit by a delegation of IMF, European Commission and European Central Bank inspectors, known as the troika, the newspaper said.
Even while both men are sick, however, the prime minister's office said Monday that Finance Minister Vassilis Rapanos had turned in a letter of resignation Prime Minister Antonis Samaras, The Wall Street Journal reported.
Samaras, 61, who is recovering from eye surgery for a detached retina and has been told by doctors not to travel, has accepted the resignation Rapanos, who was suffering from a stomach ailment.
"The troika visit will be put off for a few days, and a final date for their arrival will be decided by Tuesday or Wednesday," government spokesman Simos Kedikoglou said Sunday.
Greece will be represented at the summit by a government delegation led by Foreign Minister Dimitris Avramopoulos and acting Finance Minister Georgios Zanias.
Zanias -- left over from a two-month caretaker government after the first national election in May failed to produce a government -- was one of the chief negotiators of the original Greek bailout.
The summit had represented Samaras' first opportunity to meet with other European leaders face-to-face since Greece's June 17 elections, which he narrowly won after campaigning on a pledge to renegotiate the bailout austerity imposed by international lenders.
Greece is expected to ask for a two-year delay to 2016 on various financial targets it agreed to reach in order to keep international aid available.
Another key issue to be discussed at the summit is a "banking union," where countries would backstop one another's banks and bonds while at the same time policing one another's spending.
The Bank for International Settlements, a self-described clearinghouse "bank for central banks" whose board includes Federal Reserve Chairman Ben Bernanke, came out Sunday in favor of a banking union, including stronger measures to guarantee deposits and prevent bank runs.
The IMF and ECB have made similar pleas.
Germany has balked at the new proposals, including the eurobond and using German taxpayers' money to directly bailout banks in another country. CNBC reported Monday Germany is unlikely to agree to any new policies unless they include more centralized control of national budgets.
Merkel, meanwhile, must keep support in Parliament and the German economy from eroding further.
In a recent estimate, Barclay's said Germany's economy grew 0.2 percent in the second quarter.