Cooper was at Valley Parade for Bradford's 2-1 win over Rotherham on Tuesday and has taken his first training session.
His arrival allows David Wetherall, who has been assisting Jackson, to revert to his youth team coaching role.
Cooper, who made over 600 appearances for Millwall, Nottingham Forest and Middlesbrough, moved to the Riverside Stadium as reserve-team manager in 2006.
He was promoted to first-team coach before becoming Southgate's assistant three years later.
He also had a short spell as caretaker manager following Southgate's departure.
Source: BBC Sport
Source: BBC Sport