Social Democrat Ingibjorg Gisladottir has been asked to form a coalition
Iceland's President Olafur Ragnar Grimsson has asked the leader of the Social Democratic Alliance to try to form a new interim government.
He told reporters he had asked party leader Ingibjorg Gisladottir to hold talks with the Left Green Party on forming a minority coalition.
Conservative Prime Minister Geir Haarde announced his government's collapse on Monday, after weeks of economic crisis.
The interim government would run the country until elections in May.
The announcement came after Mr Grimsson held individual talks with Ms Gisladottir, the outgoing foreign minister, and the leader of the Left Green Party, Steingrimmur Sigfusson.
The two parties disagree on a number of issues but the president said he believed a coalition could be formed swiftly, and would be backed by the Progressive Party.
Ms Gisladottir, who recently had treatment for a brain tumour, told reporters she hoped talks to form a minority coalition with the Left Green Party would be completed before the weekend.
"I thank the president for his trust and we will try and come to a quick conclusion in our discussions," she is quoted by Reuters news agency as saying.
"We have been given this job, and we'll do our best in a difficult situation," Mr Sigfusson said, quoted by the Associated Press news agency.
The Social Democrats' outgoing social affairs minister, Johanna Sigurdardottir, has previously been tipped for prime minister by Ms Gisladottir.
The Left Green Party has soared in popularity since the country was stricken by a mounting economic crisis in October.
October 2008 - Government takes control of three largest banks
20 November - IMF approves $2.1bn (£1.4bn) loan for Iceland
26 November - Annual inflation rate hits record 17.1%
20 January 2009 - Economy forecast to shrink by 9.6% in 2009
23 January - PM Geir Haarde calls snap election for 9 May
26 January - Government resigns following breakdown of coalition
It reportedly wants elections brought forward, possibly to April.
Meanwhile, the Social Democrats want to replace the governor of the central bank, whom many blame for the country's sudden lurch from prosperity to economic meltdown.
They also reportedly want to hold a referendum on European Union membership.
"It's safe to say that no government in the history of Iceland has had to deal with as many and as complex problems as the one that is now leaving office," President Grimsson said after the collapse of Mr Haarde's government.
'Slow to react'
Iceland's financial system collapsed in October under the weight of debt, leading to a currency crisis, rising unemployment and daily protests.
The economy is forecast to shrink by almost 10% this year.
Prime Minister Geir Haarde faced daily protests
The extent of its problems became evident as conditions tightened in global credit markets last year.
It emerged that the country's banks, which had amassed debt during years of rapid expansion, owed about six times the country's economic output.
Money from around the world had also poured into Iceland because interest rates there exceeded 10%.
Mr Haarde's government responded to the financial collapse by nationalising leading banks. It also negotiated about $10bn in loans with the International Monetary Fund and donor countries.
The coalition between Mr Haarde's Independence Party and the Social Democratic Alliance had been under strain in recent months.
Ms Gisladottir has criticised Mr Haarde for being slow to react to the crisis, but he told reporters on Monday: "We couldn't accept the Social Democratic demand that they would lead the government."