By Simon Clark

The Black Lives Matter movement is reinvigorating a yearslong campaign to push some of London's oldest financial institutions to pay reparations to the descendants of slaves.

City of London companies played an important role for centuries in organizing and funding the trans-Atlantic passage of African slaves and the Caribbean and American plantations where they were forced to work. Directors of the companies earned fortunes from the trade. The Bank of England, Barclays PLC and Lloyd's of London insurance market are among those to apologize for or acknowledge links to slavery since the May killing of George Floyd by a police officer in Minneapolis prompted protests world-wide.

"Britain was the most efficient and profitable slave trader in terms of return on capital largely because of the role of the City of London in providing cheaper finance, better insurance rates, better reinsurance rates and, critically, funding the construction of the shipping industry, " Hilary Beckles, chairman of a reparations commission representing Jamaica, Barbados and 10 other Caribbean nations, said in an interview.

Mr. Floyd's death has sparked renewed calls from Caribbean governments and from Black British campaigners descended from slaves for British companies to pay reparations. They say apologizing isn't enough and are calling for more discussion about reparations. Companies are so far resisting those calls, choosing to focus on improving workplace diversity.

On July 6, the Caricom Reparations Commission, chaired by Mr. Beckles, called for British companies to participate in a summit to discuss how they can contribute to the Caribbean.

"The City of London as we know it now would not have been without the slave trade," Mr. Beckles said. "We are calling for a dialogue in which we say from our point of view this is what we think would be an appropriate attempt at remedy. In the case of the financial institutions of London we are looking for a development strategy."

The City of London Corp., a centuries-old council which manages the financial district, declined to comment on whether it would participate in a reparations summit, as did spokesmen for the Bank of England, Barclays, Lloyd's of London, Royal Bank of Scotland Group PLC, Lloyds Banking Group PLC and law firms Freshfields Bruckhaus Deringer and Farrer & Co. They have all acknowledged historic links to slavery.

British politicians have long resisted discussions on the issue. Speaking to Jamaica's Parliament in 2015 on the most-recent official Caribbean visit by a prime minister, David Cameron emphasized Britain's role in ending the slave trade. "I do hope that, as friends who have gone through so much together since those darkest of times, we can move on from this painful legacy," Mr. Cameron said.

"Finance could be the best healer," Darrel Blake, a former banker who leads a slave-trade money trail tour through London's financial district, said on a June 20 tour.

The Bank of England said it wasn't directly involved in the slave trade but "is aware of some inexcusable connections involving former governors and directors and apologizes for them," a spokeswoman said.

Lloyd's of London, the insurance market founded in the 1680s, said it was sorry for its role in the trade.

"This was an appalling and shameful period of British history, as well as our own, and we condemn the indefensible wrongdoing that occurred during this period," a spokesman said.

A Barclays spokesman said the bank is "committed to do more to further foster our culture of inclusiveness, equality and diversity, for our colleagues, and the customers and clients we serve."

Mr. Beckles said commitments to improve diversity are a public-relations stunt. "That is not a sincere effort to participate in reparatory justice," he said. "We are talking about the damage and the harm done to millions of people and the death of millions of people."

Last year he brokered a GBP20 million ($25 million) reparations agreement with Scotland's University of Glasgow, which benefited from slavery. The university is raising the money mainly through grants and donations to work with the University of the West Indies on research projects to improve health care and economic development in the Caribbean. The agreement is a model for companies to follow but no such conversations have started, Mr. Beckles said.

"We are happy to share our experience," David Duncan, the University of Glasgow's chief operating officer, said in an interview.

That agreement has set a precedent that makes companies wary, said Malik Al Nasir, a British citizen who has researched his family history and found he's descended from slaves and slave traders in Guyana.

"Companies know that there is liability here," Mr. Al Nasir said in an interview. He wants the United Nations to lead a truth and reconciliation commission to investigate and quantify slavery reparations. U.K. lawmaker Layla Moran has written to companies with links to slavery and asked them to do more.

Calls for reparations for slavery are also increasing in the U.S. in the wake of Mr. Floyd's killing, according to Ana Lucia Araujo, a Howard University professor who studies slavery. A bill embracing the idea was first introduced in Congress in 1989 but the issue of reparations has languished for decades.

The British government abolished slavery in the 1830s, paying GBP20 million -- the equivalent of billions of dollars today -- to compensate slave owners because they were deemed to have lost property. Beneficiaries are listed on a University College London website.

The government finished repaying debt used to fund the compensation payments in 2015. Compensating slave owners was unjust according to Mr. Blake, whose ancestors, like those of Mr. Beckles, were slaves.

"My taxes have gone back to the government to replace the money that was given to the slave masters that owned my family," said Mr. Blake, who worked at Barclays and HSBC Holdings PLC before becoming a teacher. "How ridiculous is that?"

Write to Simon Clark at simon.clark@wsj.com