The country’s 2,500 bank branches will start providing basic services again, such as paying cheques and opening accounts for salaries.
People will be able to withdraw €420 a week, in one go if they want, instead of the €60 a day since capital controls were imposed on 28 June.
Money transfers abroad are still forbidden, except to pay for medical services and studies.
The Athens stock exchange will remain shut until the banking and financial sector regain some stability.
“Capital controls and restrictions on withdrawals will remain in place but we are entering a new stage which we all hope will be one of normality,” the head of Greece’s banking association, Louka Katseli, said on Skai TV channel.
Katseli also called on the Greeks to bring their deposits back to the banks to “strengthen the liquidity of the economy”.
According to Kathimerini newspaper, the closing of the banks as well as capital controls cost Greece €3 billion in lost exports and commodity. The calculation does not include tourism, a key sector for the Greek economy.
Greek authorities were able to ease the restrictions after the European Central Bank (ECB) decided on Thursday (16 July) to increase the level of liquidity available to Greek banks by €900 million.
Greece’s financial short-term situation also improved after EU countries agreed on Friday (17 July) to lend Greece €7.16 billion through the EFSM rescue fund.
The loan will allow Greece to repay a €4.2 billion debt to the ECB and €2 billion to the International Monetary Fund on Monday.
The liquidity assistance freeze and capital controls were decided after Greek prime minister Alexis Tsipras announced a referendum on austerity measures asked for by Greece’s creditors.
Following a deal on a new bailout in exchange for reforms and budget cuts, Tsipras reshuffled his government on Friday (17 July) to replace critics of the agreement.
All cabinet members of Syriza’s Left Platform left the government, including their leader, energy minister Panagiotis Lafazanis.
Tsipras also appointed a special anti-corruption committee, chaired by Panagiotis Nikoloudis.