"How does Greece come out of this? In the long term, they will have to confront an exit from the euro," Grillo said on Monday.

Greece has effectively defaulted on a debt to one of its major creditors, the International Monetary Fund (IMF), and without further aid could soon default on payments due to the institution keeping Greek banks afloat right now: the European Central Bank (ECB).

Former comedian Grillo's anti-euro movement has gained traction among Italy's disaffected voters over the past few years. It has been watching the Greek debt crisis closely, as it could set a precedent for a country leaving the single currency union.

Sooner or later, debt-stricken Greece will have to "confront an exit from the euro," Beppe Grillo , the outspoken leader of Italy's anti-austerity 5-Star Movement, told CNBC -- and Italy should follow suit.

Greece is at a crossroads following a referendum on Sunday, in which 61 percent of Greeks voted "no" against the country's bailout program and the austerity measures it entails.

Following the vote, the leftwing Greek government hopes to return to discussions with lenders and euro zone partners in a stronger position to negotiate more financial aid, but with fewer spending cuts.

Whether there is any appetite in Europe to give Greece another chance – and another bailout -- is unclear, however. On Tuesday, the Greek government is expected to present new reform proposals to euro zone finance ministers and officials, who are meeting to discuss their next move.

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It comes amid concerns that letting Greece leave the euro could set a precedent for other, poorer members of the region. If it left the single currency, Greece could devalue an alternative currency and regain competitiveness.

Grillo said that Italy -- a country with the second-highest debt pile in the euro zone after Greece, according to the European Commission -- should leave the 19-member single currency union.

When asked if Italy should return to the lira, its currency before the euro, Grillo was emphatic: "Yes, with a central (bank) that has the exchange rate under control… the European Central Bank could do that. Each (country) with its own currency, and the ECB controlling fluctuations in the exchange rate."

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He conceded that he was not an economist, however, but advocated the pro-growth, anti-austerity policies suggested by economists Joseph Stiglitz and Paul Krugman.

"They're saying this system isn't working. I say it with my soul. I say with the common sense of a comedian," he said.

Despite Grillo's ambivalence towards the euro and austerity, Italy's economy is on the path to recovery after a prolonged recession. The country's economy grew 0.3 percent in the first quarter of 2015 on the previous three months, according to Italian statistics agency ISTAT.

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Although the Italian government is slowly trying to reform the labor market to make it more flexible, unemployment in Italy remains high, coming in at 12.4 percent in May, according to Eurostat.

The rate is nowhere near as bad as in Greece, however, where 25.6 percent of people were without work in March and around half of young people are unemployed.