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The Hollywood star branded the show, in which Lloyd Webber is on the judging panel, as a "13-week promotion for a musical".
Spacey, currently the artistic director of London's Old Vic theatre, added that the BBC was "crossing the line" by giving the musicals free publicity.
Lloyd Webber told Radio 1's Newsbeat: "He [Spacey] didn't actually mention my name. I know he was only talking about the shows in a generic way.
"He was really saying why can't we have a programme about plays on BBC One?"
I'd Do Anything pits contestants against each other to win a role in a new West End version of Oliver!
