Wright, 19, has been dating Bower, who plays Gellert Grindelwald in the saga's final installment, Harry Potter and the Deathly Hallows, for six months.
"We are all absolutely shocked and surprised and can't believe that Jamie has asked Bonnie to marry him," a source close to the film told the paper. "He says they are completely in love. It all seems very whirlwind, but they both say it's the real deal."
Wright confirmed the two were an item after the Bafta awards in February.