Euro rise 'cause for serious concern': Draghi

European Central Bank (ECB) President Mario Draghi said the strengthening euro was "cause for serious concern" on Thursday, and suggested the bank could take action next month after holding rates in May.

"It (the exchange rate) is very important for price stability and for growth," Draghi told journalists at his regular conference in Brussels.

"The strengthening of the exchange rate, in the context of low inflation, is cause for serious concern in the view of the (ECB) Governing Council."

For a sixth consecutive month, the ECB kept the base rate at 0.25 percent and announced no new steps to stimulate the euro zone economy.

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"The Governing Council is comfortable with acting next month," Draghi said, but it wanted to see new staff projections on the state of the economy first.

This is despite ongoing speculation that Draghi might act to combat the euro zone's disinflation problem, or address the spiraling euro.

Although inflation picked up slightly in April to 0.7 percent, it remained way off the ECB's target of close to 2 percent.

At his news conference after the policy announcement, Draghi reiterated that the ECB was "unanimous" in its willingness to use "unconventional" monetary tools to combat low inflation if necessary.

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He also emphasized that the ECB was "comfortable" with taking action when it meets again in June.

The euro rose to a two-month high against the U.S. dollar after the central bank's announcement. It continued heading toward $1.40 during the conference, before falling back. It is up around 1.1 percent on the year.

The pan-European Stoxx 600 stock index briefly reversed direction to turn lower, before heading higher again.

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Draghi also referred to events in Ukraine, saying that geopolitical risk could have a negative impact on the economic outlook, and would be carefully watched.

He added that the ongoing appreciation in the euro would be "monitored closely" by the ECB.

In a later interview on CNBC's "Squawk on the Street," former ECB President Jean-Claude Trichet said that the euro was not a target of the central bank's monetary policy, but that a strong euro during a time of low inflation was "clearly an issue."

Trichet added that despite Europe showing strong fundamentals in its economic recovery, growth and job creation have both underwhelmed central bankers. And asked why interest rates in Southern Europe have moved lower recently, Trichet attributed it to investors discovering "resiliency" in the euro.

"What we are observing is good," Trichet said. "Of course, it doesn't call for any complacency. ... The fundamentals have considerably improved in the so-called periphery countries, the countries that were under stress."

—By CNBC's Katy Barnato. CNBC's Jeff Morganteen contributed to this report.