The actress, 26, has reportedly grown increasingly close to her I Saw the Light co-star, but insiders insist their romance isn’t serious.
A source told Us Weekly magazine: ”They have been hooking up. It’s casual [but they] text when they’re apart.”
Elizabeth split from fiance Boyd Holbrook in January and isn’t looking for more than ”fun” in her personal life at the moment.
The source added: ”She wants to make a new life for herself. She’s having fun.”
Meanwhile, Tom, 34, was previously romantically linked to record producer Jane Arthy.
Elizabeth previously admitted she ”loved” being in relationships and is excited at the thought of getting married.
Speaking before her split from Boyd, she said: ”I’m relaxed about it, but I am excited and planning ideas because it’s fun – it’s just like throwing a party to celebrate life. I love being in a relationship a lot.”
The blonde beauty also admitted she was thinking about having children with her then-fiance.
She said at the time: ”My mother would love it if I was in LA with kids. I loved growing up in LA, but I always think about places I’ve never been to before, like Oregon.”