The new couple were pictured together ahead of their forthcoming KGB spy drama where they star alongside one another.
It was exclusively revealed by The Sun's Bizarre column in February that Claire had split with her husband, and father of her child, Stephen Campbell Moore.
It came a month after the actor, 38, had an operation to remove a tumour in his brain.
Stephen and new flame Sophie spent last year together filming the drama Red Joan.
The pair were pictured walking arm in arm and cosying up to one another outside a cafe in London.
Last month Claire revealed she had a "breakdown" at the age of 23.
Opening up about her life-long battle with anxiety, the now 34-year-old actress said she didn't "sleep or eat for a month".
In an interview with the Daily Star, she spoke about feeling "overwhelmed" and unable to "cope" as her career flourished.
The star said: "When I was 23, I had a breakdown. I didn’t sleep or eat for a month. It was awful.
"My life had changed. I felt overwhelmed."
Claire credited the arrival of her daughter Ivy Rose in 2015 with her recovery.
She explained: "“Having [Ivy Rose] made me think, ‘You need to sort your life out’.
"I just didn’t have to live like that. It was unbearable.
"I was sick of myself.”
Last month, Claire showed off an edgier look as she posed with her co-star Ryan Gosling at the 66th San Sebastian International Film Festival at Victoria Eugenia Theatre, in Spain.
The actress, who famously played the Queen in The Crown, turned heads in a slinky green velvet dress.
Her cropped hair was slicked off her face and she accessorised with a long necklace, while Ryan looked very dapper in a burnt orange suit and floral shirt.
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