Former Everton and Oldham midfielder Adams, who also played for the Canaries, had most recently been overseeing the club's footballers who were on loan away from Carrow Road.
Adams, who guided the club's academy team to FA Youth Cup success in 2013, had initially taken over as caretaker manager from Chris Hughton in April 2014, but was unable to keep Norwich in the Premier League.
After being appointed full-time following relegation to the Championship, Adams eventually resigned in January 2015 with the team seventh in the table, and later returned to the club on the backroom staff.
Norwich also confirmed that as part of the changes Lee Dunn will step up into the position of head of recruitment, with former midfielder Andrew Hughes appointed as the club's new loan player manager and John Iga joining as head of data and innovation.
Adams is set to help Webber "oversee and manage all football operations and strategy across the club".
Webber said on www.canaries.co.uk: "I've been fortunate enough to have worked very closely with Neil during my time with the club. He brings a wealth of experience and some great attributes to this new role.
"Neil will take on some of my responsibilities, allowing me to drive forward and progress other areas of the club."
Webber added: "It's an exciting time to be part of Norwich City, of which every player and staff member is extremely grateful for."
Norwich, promoted as Championship winners last season, currently sit bottom of the Premier League table with no points after losing all six of their Premier League games so far.