Dougray said: "I kind of feel she (Michelle) had more pressure playing Marilyn Monroe.
"She (Marilyn) was more famous than Arthur Miller, although he is an iconic role and a great inspiration to me as well because I'm an actor."
He added: "I can imagine [Michelle felt pressure] but she was amazing. She's a wonderful actress and lovely girl as well. I had great fun working with her. She'll be amazing."
The film is based on the diary of Colin Clark, who worked as an assistant on The Prince And The Showgirl in 1956, which starred Marilyn and Sir Laurence.
Dougray said: "It was great. It was a wonderful script, a wonderful director Simon Curtis, a wonderful cast.
"And Arthur Miller's such an iconic, interesting, enigmatic character, I had great fun researching that and filming that. It was wonderful."