The production, which runs at the Barbican from 6 August to 31 October 2015, caused unprecedented online queues when tickets went on sale earlier this month.
But now those who were unable to bag tickets have the hope of a big screen viewing. Cumberbatch told the Mail: "I felt very strongly that as wide an audience as possible should have access to it."
Negotiations for the screening are reportedly "well advanced".
A number of £10 tickets to the production, which is directed by Lyndsey Turner, will go on sale next year.