Almost £300 million of Britain's most cherished artworks have permanently left the country since 2011.
Stephen Deuchar, director of the Art Fund, said the export system was abused by wealthy buyers using "exploitable loopholes" and "gentlemen's agreements" to keep masterpieces out of public reach.
Among the exodus of treasures was a £50 million Picasso - Child With A Dove - now thought to be in Qatar.
Important cultural objects bought by overseas collectors are typically issued with a temporary export ban, giving British dealers or museums a chance to match the price.
Some 32 "national treasures", worth a total of about £70 million, have stayed in the UK this way since 2011-12, Arts Council figures show.
Recently, a New York-based buyer refused to sell a £30 million Pontormo painting to the National Gallery despite the institution raising the asking price.
Dr Deuchar told the Press Association this happens "time and again", adding: "Applying for an export licence you have to promise if a museum raises a matching sum you will sell it to them.
"We (the Art Fund) want to see some proper legal muscle to a system currently based on gentlemen's agreements ... the civil servants running it need to listen to people who have new ideas for change."
But leading art historian and dealer Bendor Grosvenor argued that Britain's export system is "probably the best in the world" as it balances the rights of the public and private collectors.
He said: "Art is a global business ... is it in the public interest for the state to effectively seize someone's private assets?
"Something about art brings out the socialist in everyone. We sometimes think it must belong to the people.
"Germany recently tried to tighten up their laws and there were truckloads of art leaving the country. It's a case of be careful what you wish for..."
Artworks are classed as "national treasures" if their departure from Britain would be considered a loss on one or more of three criteria: historical importance, appearance and scholarship.
Dr Grosvenor added: "There's the downstream fact that 80% of art in public collections is never on display but in storage.
"In fact, there's a sensible argument to say a work in private hands is more likely to be loaned and seen by people."
In total, 41 national treasures with a total value of £278 million were given export licences from 2011-2016.
The Department for Culture, Media and Sport said: "The UK's cultural export controls helps to keep national treasures, such as TE Lawrence's dagger and Jane Austen's ring, in the country.
"While it's not possible to save every object, the system is designed to strike the right balance between protecting our national cultural heritage and individual property rights."
The "national treasure" status is decided by the Reviewing Committee on the Export of Works of Art and Objects of Cultural Interest (RCEWA).
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