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Mackintosh wants the Cold Feet star to play the part of Fagin when he opens the new version of the musical later this year.
According to the Daily Mail, Nesbitt met the producer for lunch on Wednesday to discuss the role.
Nesbitt reportedly said he would consider the offer, but hasn't confirmed he will take the job.
Sacha Baron Cohen has already turned down the role because it conflicts with his upcoming movie projects.
The production will be linked to the BBC's latest talent show I'd Do Anything, which has launched a hunt for actors to play the roles of Oliver and Nancy.
Nesbitt was rumoured to be in line to play the next Doctor Who last year, but he later denied the reports.
