The trend is what counts, and (from a low base) here are two mildly encouraging signs for the euro (my emphasis added in the case of the Italian story)

Greece

(Reuters) – Greece’s central bank chief said on Tuesday that bank deposits were returning at a “satisfactory pace” after elections last month reduced uncertainty about the country’s future in the euro zone.

Italy

(Reuters) – Foreign deposits at Italian banks continued to drop in April at a sharp 20 percent annual pace, data from national banking body ABI showed on Thursday, confirming a trend of shrinking cross-border financing in the euro zone. Italian banks lost a net 94 billion euros in funding from abroad in the twelve months to April, ABI said. In a more encouraging sign, the data confirmed a rise in overall deposits at Italian banks highlighted this week also by Bank of Italy data. Total deposits rose 1.7 percent in May following a 1.2 percent increase in April.