The former Leeds and Liverpool forward was appointed by Crawley in May 2017 and led the club to a 14th-placed finish in Sky Bet League Two last season.
Kewell replaces Kevin Nolan after he was sacked with Notts County bottom of the Football League.
The Australian will be joined at Meadow Lane by assistant Warren Feeney, while the Magpies are on the verge of appointing a "vastly experienced" technical director.
Darren Fletcher is to stand down from his role as vice-chairman.
Kewell cut his coaching teeth at Watford (Getty Images)
Owner and chairman Alan Hardy said: "Having now worked in football for a significant period of time, I have had the opportunity to learn from other clubs what good looks like.
"I have therefore decided to appoint a technical director, who will set the club's overall footballing philosophy and be on hand to support and mentor the manager as required.
"Harry will, however, be solely responsible for our first-team squad, having the final say on transfers, tactics and team selection.
"He has enjoyed an excellent start to his managerial career and I am hugely impressed with his knowledge and attention to detail."