Not since before the global financial crisis have Europe's banks been able to obtain dollars as cheaply as they can now, a development that may puncture the euro's surprising strength.

The cost of swapping euro funding streams for those in dollars to service foreign loans fell this year to the lowest since January 2008, data compiled by Bloomberg show. The cross-currency basis swap has shrunk from a three-year high reached in 2011 as Europe's banks focused on shrinking their balance sheets instead of expanding abroad.

"Since the basis swap has now normalized, it signals the deleveraging that has been the key to the euro's strength may have largely run its course," Chris Walker, a foreign-exchange strategist at Barclays Plc in London, said in a Feb. 19 phone interview. "The drivers of the currency will return to relative monetary-policy divergence."