If speculation in the arts world is correct, the move could prove controversial because Sir Richard's report, which was commissioned by Chris Smith, the Secretary of State for Culture, Media and Sport, has not yet been completed and is not due to be published for another six weeks.
There is at this point no confirmation of the rumours which are rife in artistic circles in the wake of the resignation earlier this week of Mary Allen as chief executive of the opera house.
The Department of Culture has not been consulted on the matter, its spokeswoman said. "We would be informed of an appointment but it's not something we'd been involved with."
A spokeswoman for the Royal Opera House said: "The post has not yet been advertised". Asked whether Sir Richard was considering applying she said: "That's something he'll have to consider when the post is advertised if he feels that is appropriate."
The appointment of Ms Allen to the job was criticised by a Commons select committee precisely because, as secretary-general of the Arts Council, she was given the job in a backroom deal without the post being advertised.
At the Arts Council, which would have to approve the appointment, its press officer, Sue Rose, said: "They got into trouble with the process last time when they bounced Mary into the job. Everybody's speculating like mad but I can't believe they would make the same mistake again."
The Eyre report, which is due to be published early in May, is believed to recommend that the Royal Opera Company should be privatised but the Royal Ballet - which currently enjoys critical acclaim - continue to be funded by the Arts Council. The ROH building would be funded directly by government as a receiving house for the two companies.
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