Mirren has won rave reviews for her role as the monarch in the West End production, which dramatises pivotal meetings between the Queen and 12 British prime ministers over 60 years, and the show's director wants the royal to see the play for herself.
Daldry reveals he has reached out to the ruler about the possibility of staging the production at royal residence Windsor Castle.
He tells the London Evening Standard, "We've already offered to play the production without decor at Windsor Castle for Her Majesty. I hope that is taken up."
Daldry also heaped praise on Mirren, who won an Oscar for her turn in 2006 movie The Queen, adding, "It's incredibly challenging as an actress. It's one of the great tour de forces we've ever seen on the British stage."