The news is the latest sign of the changing climate in the art world toward the Sacklers, who are major donors to museums. Family members own Purdue Pharma, which makes the painkiller OxyContin and is facing hundreds of lawsuits as a result of the epidemic of opioid addiction.

Tate’s statement came two days after Britain’s National Portrait Gallery said it would not accept a long-discussed $1.3 million donation from the London-based Sackler Trust, one of the family’s charitable foundations. It said the decision was taken jointly by the gallery and Trust.

But the Thursday announcement, affecting Tate Modern and Tate Britain in London, as well as Tate Liverpool and Tate St. Ives in Cornwall, could have a bigger impact in the art world. All these galleries are major tourist attractions as well as home to large, high-profile exhibitions.

In an email, a spokesman for the Mortimer and Raymond Sackler family said, “We deeply sympathise with all the communities, families and individuals affected by the addiction crisis in America. The allegations made against family members in relation to this are strongly denied and will be vigorously defended in court.” He did not comment on Tate’s decision.

Much of the focus on the Sacklers’ donations to art institutions has been in the United States, where deaths and addiction associated with prescription opioids have become an unrelenting crisis. But awareness of the crisis is high in Britain, and that has led to pressure on galleries, with the news media asking frequent questions about donations from the Sacklers.