Boyle wants Firth for the role recently vacated by Michael Fassbender, according to Twitch Film. Meanwhile, the Slumdog Millionaire director hopes to cast Johansson as a character named Elizabeth.
Zoe Saldana and Mélanie Thierry are allegedly being considered for parts in the film as well.
It was previously reported that James McAvoy would be replacing Fassbender in the film. However, the website stated that this is not true. McAvoy and Fassbender were both originally cast in two separate roles in the film, and Fassbender ultimately elected to drop out due to scheduling conflicts.
Trance tells the story of a gang's efforts to find a stolen painting that disappears after their art gallery accomplice suffers a blow to the head.
Firth is currently filming Gambit opposite Cameron Diaz, Michael Caine, Alan Rickman and Stanley Tucci.
Watch the trailer for X-Men: First Class, featuring James McAvoy and Michael Fassbender, below: