A so-called "European FBI" could save the region hundreds of millions of euros, a long-standing Member of European Parliament (MEP) told CNBC, adding that a "lack of trust" between governments is stopping the idea becoming reality.

Bill Newton-Dunn, a British Liberal Democrat MEP, said the lack of cross-border cooperation between European Union (EU) member states meant criminals could avoid capture.

"Have you ever seen the movie Bonnie and Clyde? They were robbing banks in the 1930s and they got away with it for a long time. The police couldn't cross the frontier to the next state to pursue them. That is how the American FBI got started," Newton-Dunn told CNBC.

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He said Europe was in the same position today. "It has open borders and criminals are being very smart and realizing they have a good opportunity to run - all police forces are forbidden to cross a border. It is a very simple thing, if we are going to have open borders then we have to police who can patrol those borders," he said.

The EU's law enforcement agency, Europol, is primarily an intelligence-gathering body, and does not have the power to make arrests.

As such, Newton-Dunn, who was elected to the first European Parliament in 1979, called for a cross-border police force in the EU with wide-ranging powers. He said this European FBI could save the EU "hundreds of millions of euros" in tax revenues lost as a result of criminal activities, such as the sale of counterfeit goods and pharmaceuticals, and VAT fraud.