The painting will be auctioned at Christie’s this month and is expected to fetch between £11 million and £18 million, the Observer newspaper reports.
It depicts Sir David, who ran the opera house from 1945 to 1970, and was commissioned for the building in London’s Covent Garden in the 1970s.
“This was a really tough call,” Alex Beard, the ROH’s chief executive, told the Observer.
“But we have to face the situation we are in and if we can remain viable and get through this, then we can get back to employing people in the future.
“We are the biggest arts employer in the country and we knew we had to look at any assets we had. And there is only really one of any note that stands out and that is this portrait.”
The sale is part of a four-pronged plan to protect the venue’s standing as the home of the Royal Ballet and of international opera in the face of the pandemic.
That strategy also includes staff redundancies and a major drive for donations, the paper said.
