Cooper, 40, will face competition from Steven Boyer ('Hand to God'), Ben Miles ('Wolf Hall Parts One & Two'), Bill Nighy ('Skylight') and Alex Sharp ('The Curious Incident of the Dog in the Night-Time') in the category.
While Mirren, 69, will fight Geneva Carr ('Hand to God'), Elisabeth Moss ('The Heidi Chronicles'), Carey Mulligan, ('Skylight') and Ruth Wilson ('Constellations') to bag the award, said The Hollywood Reporter.
Kristin Chenoweth ('On the Twentieth Century'), Leanne Cope, ('An American in Paris'), Beth Malone ('Fun Home'), Kelli O'Hara, ('The King and I'), Chita Rivera, ('The Visit') comprise the nomination for best performance by an actress in a leading role in a musical.
The play "Wolf Hall Parts One & Two" sits at the top with eight nominations, including the best play category, best actor in a leading role, best direction for Jeremy Herrin and best costume design.
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"Hand to God" comes close to "Wolf Hall Parts One & Two" with five nominations. Besides the best actor and actress category, it is also nominated for the best direction for Moritz von Stuelpnagel and in the best actress in a featured role category for Sarah Stiles.
The nominations for the best musical includes "Fun Home", "An American in Paris", "Something Rotten!" and "The Visit".
Other nominations in the best direction of a play category include Stephen Daldry ('Skylight'), Marianne Elliott ('The Curious Incident of the Dog in the Night-Time'), Scott Ellis, ('You Can't Take It with You').
Tommy Tune will receive the special Tony Award for lifetime achievement in the theatre.
The awards are set to be handed out June 7 in a ceremony, to be hosted by Kristin Chenoweth and Alan Cumming.