Iglesias sounds similar to his close friend Alexis Tsipras, Greece's new prime minister: "Austerity measures are destroying Europe," he told CNBC. "As a pro-European, I think we are in a situation in which we must rectify."

Even more startling, the most recent polls for the general election coming this autumn show Podemos with a small lead over the two long-standing establishment parties, an indication of just how much resonance the party's "anti-austerity" platform has with the Spanish population.

Podemos (Spanish for "We Can") cemented itself in the Spanish political scene by garnering more than a million votes in May elections for the European Parliament, where the party now holds five seats. The accomplishment is all the more extraordinary for coming from a political party little more than a year old.

That's a fact that looms large among European finance ministers as they continue their bailout negotiations with Greece . According to multiple sources who spoke with CNBC, finance ministers and prime ministers on the continent express concern that if they give leniency to Greece, they will embolden other anti-austerity movements. Iglesias' party, Podemos, is at the top of the list.

Pablo Iglesias is just 36 years old, and yet, he could become the next prime minister of Spain .

Pablo Iglesias, leader of the Spanish Podemos (We Can) left-wing party, applauds before giving a speech to his supporters in Madrid, Spain, Jan. 31, 2015.

Iglesias is diminutive in stature, but he looms large in the minds of other European leaders as he rails against the very reforms they think will make Spain and the rest of Europe more competitive.

He wants to raise the Spanish minimum wage and lower the country's retirement age, a step that he believes will reduce youth unemployment. Restructuring the county's debt is also on his list, although such a step would be highly likely to scare off creditors and lead to higher interest rates in Spain.

Read MoreIs the Greek PM on a collision course with Europe?

In an interview with CNBC at the New York Stock Exchange, he wanted to highlight greater state intervention in banking.

"There is a part of the financial system that must undertake social functions, and that implies that there must be mechanisms of public control," he said. "There must be devices that assure that there will be credit for small- and medium-size businesses and families. From there, it is fundamental to use the public sector."

Iglesias indicated a certain degree of disdain for financial institutions: "Sometimes, banks are like small children. They ask for too many things. They want five dishes for dinner. And the state has to be like a father that says no. One dish is enough and after that you must go to bed."

He was in Athens on the night Tsipras was elected, and he said he sees the ongoing Greek negotiations as a key test for Europe.



Read MoreHere's Greece's bailout extension request letter



"If they don't reach a deal with Greece, if they keep on applying a dogmatism that have taken us Europeans to a disaster, maybe in a few months they'll have to negotiate with Marine Le Pen. And I don't want that situation to happen," making reference to the leader of France's right-wing Front National party.



Asked if he would be ready to lead a major European nation at such a young age, he said, "It's a difficult question. I think that we need young people doing things. I think we have (many) old people with very old ideas. I think we need new ideas and new people ruling the government."

