LONDON >> Steven Woolfe, a British member of the European Parliament, concedes that he should have checked his facts before setting off a social media campaign against one of Britain’s leading supermarket chains.

But when Woolfe, who wants Britain to leave the European Union, last year mistakenly accused the chain, Sainsbury’s, of bankrolling those who want to remain in the bloc — using the hashtag #ShameOnYouSainsburys — the company quickly sprang into action.

Within hours it had contacted his office, insisting that it was not funding either side in the debate, stating that it had no plans to do so and requesting that he correct his message.

Woolfe, a member of the UK Independence Party, did just that, but also drew a lesson — one that could have a big impact ahead of a British referendum on June 23 on whether to leave the bloc. Polls suggest the vote could be close.

“I think that was a wonderful piece of true consumer democracy,” he said, recalling the episode. He added that companies must appreciate that they face a backlash if they speak out on the issue of British membership of the union.

Many seem to have gotten his point.

Concerns about taking sides on this divisive issue are prompting a significant number of high-profile companies to lie low. They worry that expressing any opinion about staying in the bloc or leaving could lead to backlash from customers or shareholders who hold the opposing view, or even split their own boardrooms.

That reticence is creating a challenge for the government of Prime Minister David Cameron and other groups campaigning to keep Britain inside the bloc. They had hoped that unambiguous support from businesses would highlight the economic risks to Britain of breaking away from the European Union and help persuade wavering voters to oppose withdrawal.

Many British companies have a direct interest in staying, particularly if they import from or export to the bloc’s single market of around 500 million people, to which Britain has automatic access. Multinational businesses that have made Britain an important hub for their European and global operations are similarly concerned. Other companies fret that a vote to leave would destabilize financial markets.

Yet so far, the voice of business has been less full-throated than many analysts expected.

“Business does seem strangely muted on this,” said Simon Tilford, deputy director of the Center for European Reform, a research institute in London. “It has been surprising that so few internationally active businesses are prepared to speak out.”

In February, more than a third of the companies in the FTSE 100, an index of the largest of the nation’s public companies, signed a letter declaring that Britain was better in the European Union. It was a sizable number but not the overwhelming majority some had expected.

Meanwhile, the campaign in favor of leaving has produced a list of around 250 business supporters, including some well-known figures, though no current FTSE 100 chief executives or chairmen.

Senior players in London’s big international financial companies — including Goldman Sachs International, Citigroup and J.P. Morgan — have made it clear that they want to stay. But several of Britain’s retail banks have been much more cautious, aware that they risk annoying a large number of customers regardless of what they say.

There are legal constraints, too. Electoral law prevents companies from spending more than 10,000 pounds — the equivalent of about $15,000 — to influence the result during a referendum campaign, unless they formally register as advocates.

Tilford said many companies still either thought that there was relatively little prospect of Britain’s voting for withdrawal, or believed — in his view over-optimistically — that little would change in the event of a British departure. He expects more corporations to speak out closer to the referendum, particularly if opinion polls continue to predict a close result.

In the meantime, with surveys showing voters to be deeply divided, companies appear to be worried about annoying consumers, Tilford said. The feeling, he added, is “there is no point putting your head above the parapet for something that isn’t going to happen.”

That is good news for those campaigning for Britain to leave the bloc who want to show that companies that speak out against that view may pay a price with disgruntled customers.

Woolfe said he hoped to organize social media campaigns challenging high-profile companies that have warned against British withdrawal — undeterred by last year’s confusion of Sainsbury’s (which arose after reports that a former chairman, Lord David Sainsbury, would support the campaign to keep Britain in the European Union).

In Woolfe’s sights now are two airlines: EasyJet and Ryanair. Carolyn McCall, chief executive of EasyJet, a discount airline, suggested in February that a vote to leave could herald a return to the days when flying was “reserved for the elite.” Ryanair’s chief executive, Michael O’Leary (an Irish, rather than a British, citizen), has promised to “bore everybody to death” by repeating a pro-European Union message.

Woolfe also highlighted the Virgin Group, whose founder, Richard Branson, has warned that leaving the union would be “the worst decision the British public ever made,” and the German automaker BMW, which warned its British workers that leaving the bloc would drive up costs and prices and could affect employment.

“For me, that’s an important consideration,” said Woolfe, who speaks for the UK Independence Party on migration and financial issues. “Do I buy a Japanese car, who’ve said they are going to stay and have their plant in the north of England, and not interfering in the EU referendum? Or do I go and buy a BMW, who have said they might move their jobs away if we leave the EU?”

There is a comparison here to Scotland, which held a referendum on independence in 2014. The parallel is not precise because, with the Scottish government leading the campaign to break away from the United Kingdom, businesses that argued publicly for the status quo were risking their relationship with the government in Edinburgh. By contrast, in the current European debate, the British government supports remaining in the bloc, as do the Scottish government and Britain’s opposition Labour Party.

Among the companies unlikely to break their silence is the supermarket chain mistakenly targeted by Woolfe last year.

“Sainsbury’s is a nonpolitical organization,” the company said in a statement, “and we believe it’s for the British people to decide whether we remain in the EU.”

© 2016 The New York Times Company