Sources told ESPN the deal is for six years.
Drew, 41, has been the coach of the Crusaders for the past five seasons.
"No Vanderbilt team has ever made it to the Final Four, and we would like to be that first," Drew said
He has a 124-49 record and has taken Valpo to a pair of NCAA tournament appearances. Drew's Crusaders finished 28-10 this season after falling in the NIT championship game to George Washington.
Stallings left after a 19-14 season and an opening NCAA tournament loss to Wichita State. In his 17 seasons, Stallings had seven NCAA tournament berths and reached the Sweet 16 twice. This season's berth was his first since Vanderbilt won the SEC tournament in 2012.
"We definitely want to be nationally known on a yearly basis on the basketball level," Drew said.
That's exactly what athletic director David Williams wanted when he went looking for a new coach to replace Stallings. Williams used former Vanderbilt coach Eddie Fogler as a consultant, and the athletic director said they had five serious interviews before choosing Drew with his mission of making Vanderbilt a regular in the NCAA tournament.
"You can't win the national championship if you don't play in the tournament," Williams said. "We just saw a great game: North Carolina and Villanova. I want to be North Carolina and Villanova. I want to be Villanova, so that's the expectations. Our expectations are the same: to win. But do it the right way."
As a player, Bryce Drew hit the famous buzzer-beating 3-pointer that helped the 13th-seeded Crusaders upset Ole Miss in the first round of the 1998 NCAA tournament.
Williams said Drew understands what Vanderbilt officials want.
"He wants it as bad as we do," the Vanderbilt athletic director said of his new coach.
The Associated Press contributed to this report.