But capital controls will remain in place, with Greeks only allowed to withdraw 88 dollars a day.
New sales tax rates that were agreed between Greece and its international creditors last week in return for a three-year bailout will also come into effect.
Merkel ally urges German MPs who voted against Greek aid talks to rethink
The general secretary of Chancellor Angela Merkel's Christian Democrats has urged lawmakers who voted against opening talks on a third financial rescue for Greece to reconsider.
German lawmakers on Friday gave Berlin approval to negotiate a third bailout for Greece by 439 votes to 119, with 40 abstentions. There was a sizeable rebellion among Merkel's conservatives, with nearly a fifth voting "no".
The Bundestag will have to vote again on any agreement between Greece and its creditors for a third bailout.
In an advance copy of an interview to be published in newspaper Tagessspiegel on Sunday, Peter Tauber said the suffering of the Greek people "doesn't leave any of us cold" but the examples of Spain, Portugal, Cyprus and Ireland showed it was possible to successfully overcome the debt crisis.
"So, after successful negotiations with Greece, there can be reasons for a 'yes', even for those who have said 'no' up until now," Tauber said.
He said a Greek exit from the euro zone was off the cards at the moment, but warned: "Europe has not given Greece a blank cheque, there are very clear expectations and if Greece doesn't fulfil them, then we'll sit back down at the table very quickly."
In a German radio interview Volker Kauder, parliamentary leader of Merkel's conservatives, rejected criticism of Finance Minister Wolfgang Schaeuble's suggestion that Greece might be better off taking a time-out from the euro zone.
Schaeuble made clear that such an option would only have been possible if Greece gave it the green light but Athens would never do that, Kauder said in an interview due to be broadcast on Deutschlandfunk on Sunday.
Kauder said people should therefore now concentrate on the negotiations with Greece instead as these were "hard enough".
Now that initial decisions have been taken in Greece, the question is whether Athens will actually carry out the reforms required for a bailout, Kauder said.
"They need to be done now. It really is the eleventh hour for Greece in terms of actually implementing these reforms," he said, naming the big public sector, lack of foundation for economic development and lack of tax administration as examples.
Kauder said all countries in Europe needed to become competitive and Greece must make a contribution to this, adding that even under previous governments Athens had not been willing to make the country as competitive as was necessary in Europe.
He rejected calls for a debt "haircut" for Greece, saying Germany's conservative bloc shared Schaeuble's view that such demands had to be rejected.
German Finance Minister prepared to resign
German Finance Minister Wolfgang Schaeuble says he's prepared to resign rather than go against his convictions in negotiations on Greece's debt crisis.
In an interview with Spiegel magazine, he admitted there were "differences of opinion" between himself and Chancellor Angela Merkel.
But he also said they were able to count on each other.
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