Germany, the biggest European contributor to bailout programs that have kept Greece afloat for five years, insisted it still was not too late for Athens to come to terms with its creditors at the EU and IMF.
Greeks pulled more than 1 billion euros out of their banks in a single day, banking sources said on Friday.
The European Central Bank on Friday raised the ceiling on emergency liquidity that Greek banks can draw from the country's central bank for the second time this week, a banking source told Reuters on Friday.
The report of the ECB's liquidity decision was welcomed by traders but had little effect on the euro.
"That's helpful, but it's just a Band-Aid," said Richard Franulovich, senior currency strategist at Westpac in New York.
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The euro was last flat against the dollar, after sinking under $1.13 to a session low of $1.1294.
Versus sterling, which strategists say is something of a safe-haven, the euro fell as low as 71.22 pence, its weakest since May 28. It was last down 0.01 percent to 71.50 pence.
The euro was also lower against the Swiss franc and yen, both traditionally safe plays.
The dollar had been trading weakly since Wednesday, when Federal Reserve policymakers dulled expectations of imminent hikes in U.S. interest rates, but rose modestly on Friday.
The dollar index was up 0.04 percent after touching its lowest in a month on Thursday. The dollar was down 0.25 percent against the yen.
"Greece remains highly combustible, and that's a recipe for dollar strength and risk aversion," Westpac's Franulovich said.
The euro has so far proved surprisingly resilient to worries over Greece. Although it was weaker against the dollar on Friday, it was still on track for a third consecutive week of gains.
That comes despite weeks of talks between Greece and its creditors as they try to reach a cash-for-reforms deal for the country in order to avoid a "Grexit."