As Greeks debate the fresh austerity measures that are a prerequisite for another bailout, the country’s world-leading shipping industry is coming under increasing pressure to carry more of the load.

At the behest of its international creditors, the Syriza-led government has agreed to raise taxes on the long-protected sector—one so much a part of Greek culture that it has tax breaks enshrined in the constitution.

The industry is dominated by a small circle of family-run companies that control almost a fifth of the world’s shipping fleet—long a source of national pride.

Yet recent years of economic pain have fed a deep-seated suspicion among many Greeks that the country’s high-living shipping magnates haven’t been paying their fair share.

“The country’s shipping community must be ready to lift the heaviest of burdens to help the country out of economic crisis,” Merchant Marine Minister Thodoris Dritsas told a gathering of owners in January, shortly after the left-wing Syriza party took office.

But industry executives and supporters say higher taxes threaten to drive away a business that employs more than 200,000 people and contributes around 7.5% of Greece’s gross domestic product. By comparison, tourism directly contributes about 600,000 jobs and 9% of GDP.

As part of early negotiations with international creditors for a third bailout, Athens has agreed to increase the tonnage tax—a flat, annual rate, based on a ship’s capacity, that is now roughly harmonized across the European Union. Greece also would gradually abolish some tax benefits that other EU countries also offer.

Details haven’t been hammered out. But shipowners say that if enacted, Greece would become one of the most expensive countries in the EU, in terms of tax, to own a ship—and cause many owners to weigh anchor.

“If there is a change in the tax and legal framework under which shipping operates in Greece, I would certainly expect a mass exodus by owners. That’s what I will do,” said Michael Bodouroglou, who runs a fleet of 25 dry-bulk and container vessels through two New York-listed companies, Box Ships and Paragon Shipping.

Government officials have said they won’t comment until a decision is reached, after talks with shipowners.

The prospect of higher taxes comes at a particularly bad time for the global shipping trade. It is currently being convulsed by low freight rates, steep competition on pricing and a glut of ships plying the world’s oceans.

Nevertheless, shipping is one of the few areas in Greece that is still hiring, at good wages, in a country where overall unemployment tops 25%.

According to the Union of Greek Shipowners, an industry group, revenue generated by Greek shipping totals about €13 billion ($14.1 billion) annually. Most of the companies are privately held and don’t report results.

Over past three years, the industry paid tonnage taxes of more than €100 million annually, on average, a Greek government official said.

In 2013, under government pressure to do more, Greek shipowners agreed to basically double that for four years—contributing an additional €420 million through 2017.

Counting those “voluntary” payments, “the Greeks now pay the highest tonnage tax in Europe,” said Basil Karatzas, a New York-based maritime adviser who works with some of the world’s biggest shipping companies.

For Greece’s creditors, the industry is seen as one of just a few potentially ripe sources of new government revenue that could help Athens balance its books and, eventually, pay down debt.

“The Greek shipping industry is very successful and should contribute more to national coffers,” an EU official said.

George Xyradakis, an Athens-based shipping consultant who also advises China Development Bank on ship financing, said such an assessment was based on “a misconception” by the creditors.

“What they see is that while Greece is in a terrible economic state, its shipowners invest billions every year into new vessels. So instinctively they say, these people are making a lot of money and must contribute more to national coffers,” he said.

Some high-profile scandals during Greece’s economic crisis have also touched the industry. In 2013, Victor Restis, who controls one of Greece’s biggest fleets, was arrested for alleged embezzlement from a bank his family controlled. He gave back the money and was released pending trial. He has denied wrongdoing.

The jet-setting lifestyle of some of the owners—perhaps epitomized by the late Aristotle Onassis—hasn’t helped their image amid the downturn.

In 2010, Leo Patitsas, the heir to one of Greece’s biggest shipping families, drew a rebuke from the then-defense minister after hosting a lavish wedding party aboard the battleship Averof, the Greek navy’s flagship during much of the 20th century and now a naval museum.

The Union of Greek Shipowners president, Theodore Veniamis, declined to comment on the industry’s image.

Mr. Bodouroglou of Box Ships and Paragon Shipping acknowledged the industry’s difficult position. “We are up against years of mistrust from the Greeks, personified in the new government,” he said.