Talking after the meeting, Mrs Merkel called for an end to market speculation that Greece would default and said that Germany would offer support and expertise.
"Greece and Germany will work together toward a further modernisation of Greece," she said.
She also remarked that Germany and other European Union countries have also had to implement austerity measures amid the global economic crisis.
Previous reports of potential support for Greece had proved unpopular in Germany where many people do not support their taxes being used for bailouts.
Economy Minister Rainer Bruederle said that his government "does not intend to give a cent" to Greece in financial aid.
There are also fears that rescuing one country could encourage others to expect the same.
Meanwhile, Germany passed its budget for 2010, with borrowing set to soar this year.
New borrowing is expected to reach 80.2bn euros ($109bn; £72.5bn) - double the previous highest debt record, set in 1996. However this is less than the 85.8bn euros initially proposed by the government.
On Thursday, the Greek government went to the financial markets to borrow money and saw its 5bn euro ($6.8bn; £4.5bn) bond issue oversubscribed.
Commenting on the move Mrs Merkel welcomed the uptake. "The placement of the bond yesterday went very well and that is of course a good signal for the markets," she said.
But Greece will need to borrow more in the coming months - more than $70bn for the year as a whole.
Mr Papandreou has suggested that Greece might go to the International Monetary Fund (IMF) for help.
But the other countries in the eurozone would not welcome what would be seen as a sign that they could not fix their own problems.
The president of the European Central Bank, Jean-Claude Trichet, has dismissed the idea of the IMF providing financial aid for Greece.
"I do not trust that it would be appropriate to have the introduction of the IMF as a supplier of help through standby or through any kind of such help," he told reporters in Frankfurt on Thursday.
Mr Papandreou will also meet French President Nicolas Sarkozy in Paris on Sunday before travelling to Washington to meet US President Barack Obama on Tuesday.
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