"The economic prosecutor has asked me to request it again from the French government," Stournaras told Greece's parliament, referring to a list containing details of some 2,000 Greek account holders.
He added he had sent the request and is "expecting" an answer.
The list was originally acquired from France's then finance minister Christine Lagarde two years ago and is part of account data leaked by an HSBC bank employee in Switzerland.
Mystery has long surrounded the so-called "Lagarde list", with Greek officials claiming in early October that a copy of it had been found after the original version apparently went missing.
The first recipient of the data, former finance minister George Papaconstantinou, on Wednesday told a parliamentary committee that he did not know what had happened to the original version.
Greek officials though have long claimed that the information is unusable, because it had been illegally acquired. Deputy finance minister George Mavraganis recently called the list "industrial espionage".
Increasing anger over an expected new round of austerity cuts, imposed by Greece's international creditors, has put pressure on the government to seek the list and use it to crack down on potential tax dodgers.
Elected in June, conservative Prime Minister Antonis Samaras has vowed to combat tax evasion, a chronic problem in the Greek economy.