However, the German government says the issue was resolved legally years ago.
Greece`s radical left Syriza government is making the claim while struggling to meet massive debt repayment deadlines.
Reacting to the Greek claim, German Economy Minister Sigmar Gabriel said it was "dumb" to link Greece`s bailout by the eurozone with the question of war reparations.
"To be honest I think it`s dumb. I think that it doesn`t move us forward one millimetre on the question of stabilising Greece," he said.
He said ordinary Greek citizens however deserved "huge respect" for their economic sacrifices under the bailout programme. The Greek elite had "plundered" the country, he complained.
`Legally closed`
Greek Prime Minister Alexis Tsipras raised the reparations issue when he met German Chancellor Angela Merkel in Berlin last month.
The new figure given by Greek Deputy Finance Minister Dimitris Mardas includes €10.3bn for an occupation loan that the Nazis forced the Bank of Greece to pay.
"According to our calculations, the debt linked to German reparations is 278.7bn euros," Mr Mardas told a parliamentary committee investigating responsibility for Greece`s debt crisis.
Mr Mardas said the reparations calculation had been made by Greece`s state general accounting office.
Berlin paid 115m Deutschmarks to Athens in 1960 in compensation - a fraction of the Greek demand. Greece says it did not cover payments for damaged infrastructure, war crimes and the return of the forced loan.
Germany insists the reparations issue was settled in 1990, before Germany reunified.
The budget spokesman for Chancellor Merkel`s Christian Democrats, Eckhardt Rehberg, reiterated on Tuesday that "the reparations issue is for us closed, politically and legally - the same applies to the so-called forced loan".
Syriza politicians have frequently blamed Germany for Greek citizens` hardship under the austerity imposed by international lenders.
Mr Tsipras is trying to renegotiate the €240bn EU-IMF bailout that saved Greece from bankruptcy. Greece has not received bailout funds since August last year, as the lenders are dissatisfied with the pace of Greek reforms.
A Greek repayment of €448m to the International Monetary Fund is due this Thursday.
Greek Finance Minister Yanis Varoufakis has said that Greece "intends to meet all obligations to all its creditors, ad infinitum".
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