LONDON—U.K. Prime Minister Theresa May said Britain would introduce legislation similar to the U.S. Magnitsky Act as part of its broadside against Russia, marking another potential victory for supporters of Russian lawyer and whistleblower Sergei Magnitsky, who died in prison in 2009.

The U.S. law, passed in 2012, permitted U.S. officials to freeze the assets and ban the entry of Russians believed to be implicated in Mr. Magnitsky’s death and in human-rights violations more broadly. It was expanded in 2014 to target human-rights...