Greece looks virtually certain to miss a debt repayment to the IMF on Tuesday as the country's European partners shut the door on extending a credit lifeline after Athens announced a referendum on bailout terms.
The Swiss franc, which tends to appreciate at times of market uncertainty, strengthened to 1.0315 EURCHF= as the euro fell sharply across the board in Asian trading. But by 0810 GMT the franc was just a third of a percent higher on the day at 1.0395.
SNB chief Thomas Jordan said the central bank had intervened. Jordan said last week that the franc was considerably overvalued, and a Greek default would likely spark massive safe-haven flows into the Swiss currency.
"The Swiss authorities want the euro/Swiss back up at 1.30, let alone where we are now, and let alone at 1.20," said Neil Mellor, a currency strategist at Bank of New York Mellon in London.
The franc's modest gains contrasted with the yen, the other popular safe-haven currency, which was up 1.6 percent against the euro at 136.125 EURJPY=EBS. It had earlier hit a four-week high of 133.80 yen as investors fled to safety.
Against the dollar, the euro had initially fallen to as low as $1.0956 EUR=, but it regained ground in European trade to $1.1086. That still left it 0.7 percent lower on the day.
"There is uncertainty over whether the Grexit risks implied by the gap lower in euro/dollar at the start of the Asian session will actually be realized, with the Greek referendum still just under a week away," said Stephen Gallo, European head of FX strategy at BMO Capital Markets in London.
"Contagion is definitely apparent but does seem reasonably limited so far."
The impact of the Greece crisis has been less clear in currency than in other markets, with the euro's status as a funding currency muddying the waters.
Ian Stannard, European head of FX strategy at Morgan Stanley in London, said the impact on the euro from the Greek drama would be felt acutely only if investors moved their money out of Europe entirely.
Nevertheless, the cost of hedging against sharp swings in the euro against the dollar over the next week jumped to its highest in over five years on Monday, as Greece flirted with default and a possible exit from the euro zone.
Given relatively low liquidity as investors cut their euro positions, Bart Wakabayashi, head of foreign exchange for State Street Global Markets in Tokyo, said the euro's drop so far did not suggest any panic selling in the foreign exchange market.
"It's been surprisingly orderly, as the reaction was expected because of the headlines over the weekend. It could have been much uglier," he said.