It came at the Old Bailey after the star told jurors the press would follow him about even when he made supposedly secret arrangements.
He told jurors that media attention intensified following his starring role in 2001's The Talented Mr Ripley, for which he won an Oscar nomination.
Mr Law, 41, said: "There seemed to be an unhealthy amount of information that people, or someone, had that meant they had access to my life and my whereabouts."
Later, prosecutor Andrew Edis QC asked him: "When was the first time that you heard the suggestion that a member of your family had passed information to the News of the World for money?"
Mr Law replied: "Today."
The actor was quizzed by then-NotW editor Andy Coulson's lawyer, Timothy Langdale QC, about a story about his then girlfriend Sienna Miller having an affair with actor Daniel Craig.
The lawyer wrote down a name of a source on a piece of paper to show the witness. The name was not read in court.
Mr Law told the jury that he had discovered last autumn that the relative had passed on information, but had never heard claims they had been paid. He said: "I was made aware very recently that there had been some kind of communication with this person and several others in and around and about this period of time. I was never aware any money had been exchanged."
Mr Law said he first became aware of rumours of the affair around the time the couple went to Ms Miller's sister Savannah's wedding and Mr Law was shown News of the World articles about the allegation. He said he phoned Mr Craig to confront him about the affair.