The Italy midfielder is set to undergo a medical and sign a one-and-a-half-year deal.
Currently in 8th place in the Championship, Millwall will benefit from Burke's expertise, who previously had a successful loan stint with the club. At 25-years-old, Burke has made 15 appearances for Bremen this season and has scored two goals.
The 21-year-old, who has made seven senior appearances for the Irons mostly in Europe, had failed to battle his way to becoming a first-team regular behind fellow right-backs behind Ben Johnson, Vladimir Coufal and the versatile Thilo Kehrer. Ashby's contract was due to expire in six months and West Ham had been unsuccessful in getting him to sign a new deal. Newcastle manager Eddie Howe said they had signed the defender, who is the son of former Watford, Brentford and Gillingham defender Barry Ashby, with a long-term view. "Harrison is a very talented young player with a lot of potential so we are delighted to sign him and are are looking forward to working with him in this exciting new chapter of his career," he said.