The decree sets a new cumulative weekly withdrawal limit of 420 euros ($458), with the daily limit remaining at 60 euros.
To facilitate Greeks studying abroad the decree permits the electronic transfer of up to 5,000 euros per trimester.
Meanwhile, people receiving health treatment abroad will be able to access up to 2,000 euros in cash.
A broad range of capital controls still remain in place, however, including a block on capital transfers and a ban on the opening of new accounts and addition of new depositors to existing accounts.
The decree also enables the Bank of Greece to restrict the amount of euros or other currency carried in cash out of Greece.
New sales tax rates that were agreed between Greece and its international creditors in return for athree-year bailout last week will also come into effect on Monday.
The bank closure was enacted on June 29, after the radical government of Alexis Tsipras called a referendum on lenders’ austerity demands that Greeks rejected by over 61 percent.
The three-week shutdown has cost the country’s struggling economy some 3.0 billion euros ($3.3 billion) not counting lost tourism revenue, Kathimerini daily estimated Saturday.