The Sun writes that Shaw, 45, recently moved into 31-year-old Burrows's luxury flat in Clerkenwell, London, and the couple are said to be deeply in love with each other.
An insider tells the publication, "They are a couple in love. They are so happy and they know they have a future.
"They have been together for a while but the relationship has moved up a gear and they want to commit to each other.
"Moving in together is their way of saying it's forever."
Shaw plays AUNT PETUNIA in the Harry Potter films. Burrows is best known for her appearances in a number of Hollywood blockbusters including Troy, DEEP BLUE SEA and GANGSTER NO 1.11/11/2004 02:33
Set in the beautiful Swiss Alps, Youth sees Michael Caine & Harvey Keitel in a fine piece of work.