University president M. David Rudd said Thursday on Twitter that the school "hired a great football coach today" and will hold a news conference Friday afternoon.
Norvell, 34, replaces Justin Fuente, who was hired by Virginia Tech on Sunday after four years at Memphis. Fuente came to Memphis after a stint as TCU's co-offensive coordinator.
Norvell will take over a program in much better shape than what Fuente found. The Tigers had gone 5-31 during the three years before Fuente was hired. They went 10-3 in Fuente's third season, won a bowl game and finished No. 25 in the Associated Press Top 25 for the first end-of-season ranking in Memphis' history.
Fuente left the Tigers with a 9-3 record and bowl-bound again, although he won't coach them in that game. The Tigers averaged 510.4 yards and scored 42.6 points per game.
Norvell might have to replace redshirt junior Paxton Lynch, the quarterback who threw for 3,670 yards with 28 touchdowns and only three interceptions this season.
Memphis also is working on an indoor practice facility, and local officials are talking about further improvements to the Liberty Bowl Memorial Stadium.
Norvell has run the Sun Devils' offense for the past four years under coach Todd Graham. He was promoted to deputy head coach in December 2013. This season, Arizona State (6-6) averaged 474 yards and 33.8 points per game.
He started coaching as an offensive graduate assistant at Central Arkansas, where he started all four seasons and was the school's career receptions leader.
He joined Graham at Tulsa as a graduate offensive assistant and receivers coach, and had worked only for Graham since then, following the coach from Tulsa to Pittsburgh to Arizona State.
Information from The Associated Press was used in this report.