Schottenheimer, son of NFL coaching icon Marty Schottenheimer, was the OC with the New York Jets from 2006 to 2011 before joining the Rams in 2012.
He's drawn mixed reviews as he's helped the Rams improve in scoring in each of his two seasons. The Rams finished 30th in total yards and 22nd in scoring in 2013 despite playing more than half the season without starting quarterback Sam Bradford.
The Vandy vacancy is a result of the departure of James Franklin, who replaced Bill O'Brien as coach at Penn State last week. Franklin guided the Commodores to a 24-15 mark in three seasons and won back-to-back bowl games, a first in program history.
Athletic director David Williams said Saturday that "there's already a list of 20-25 people who've reached out to us."
Stanford defensive coordinator Derek Mason, Indianapolis Colts offensive coordinator Pep Hamilton and Vanderbilt offensive line coach Herb Hand are also among the candidates, sources have told Schefter and ESPN's Joe Schad.
Vanderbilt's last search for a coach was for a program that had played in only four bowls in its history and went 36-114 from 2000 to 2010. The Commodores posted only one winning record (2008) in the 28 seasons before Franklin was hired.
Information from ESPN.com Rams reporter Nick Wagoner and The Associated Press is included in this report.