The 35-year-old actor has started shooting for the period drama in Scotland, according to The Hollwood Reporter.
The film will be directed by Annabel Jankel.
The story is set in a small town of 1950s Britain that discovers the romantic relationship between a single mother (Grainger), who is struggling with the breakdown of her marriage, and a doctor (Paquin).
The movie is based on Fiona Shaw's 2009 novel of the same name.
It also features Kate Dickie, Emun Elliott, Steven Robertson, Lauren Lyle and debutant Gregor Selkirk.