"Following the reports in today's papers, I just want to say I am deeply ashamed and upset that I've hurt Sienna and the people most close to us," Law said in a statement released Monday to the British Press Association.
"I want to publicly apologize to Sienna and our respective families for the pain that I have caused."
He added: "There is no defense for my actions which I sincerely regret and I ask that you respect our privacy at this very difficult time."
More on Jude Law
British tabloids recently reported an affair between Law, 32, and Wright, 26. According to stories, the two met in New Orleans earlier this year, while he was filming All the King's Men, when Wright, who was hired by Law's ex-wife Sadie Frost, was said to be looking after one of his children.
Law – named PEOPLE's Sexiest Man Alive last year – and Frost had been married from 1997 to 2003.
Law and Miller, 23, who became engaged last Christmas, met on the set of the movie Alfie.