The former Barnsley and Blackburn midfielder has recently led Boro's reserves to a second successive league two title but Mowbray admits he could not deny Agnew an opportunity to coach at the next level.
"I took a call from Steve Bruce earlier this week and he asked if he could talk to Steve. Aggers has an ambition to work at first-team level and I wouldn't stand in his way," Mowbray told the club's official website.
"We're disappointed because he's a very good coach and he worked extremely hard with a young group last year.
"It will be a loss to us and yet we wish him well. He's a good man and it's a loss to us but a positive for Steve and I'm sure he'll do well."