Schiano, Urban Meyer's defensive coordinator at Ohio State, and Tom Bradley, defensive coordinator under recently fired UCLA coach Jim Mora, were both Penn State assistants under Joe Paterno during Sandusky's tenure.
Both have denied allegations of their knowledge of child abuse by Sandusky from a 2015 deposition by ex-Penn State assistant Mike McQueary. The allegations became public in July 2016, at which time both Meyer and Mora defended their assistants' integrity.
"The head football coach at the University of Tennessee is the highest-paid state employee,'' Tennessee State Rep. Jeremy Faison said. "They're the face of our state. We don't need a man who has that type of potential reproach in their life as the highest-paid state employee. It's egregious to the people and it's wrong to the taxpayers.''
Faison was one of at least three state representatives who went on Twitter or issued statements to criticize the possibility of Tennessee hiring Schiano. Republican gubernatorial candidates Mae Beavers, Diane Black, Beth Harwell and Bill Lee also relayed their objections to a Schiano hire.
State Rep. Eddie Smith tweeted that "a Greg Schiano hire would be anathema to all that our University and our community stand for.''
Our Tennessee standards mean something, and a Greg Schiano hire would be anathema to all that our University and our community stand for. I sincerely hope that these rumors are not true, because even serious consideration would be unacceptable.— Eddie Smith (@RepEddieSmith) November 26, 2017
Earlier Sunday, a source confirmed to ESPN that Tennessee was finalizing a contract with Schiano.
The news was first reported by USA Today Sports.
Schiano, 51, has spent the past two seasons as Meyer's assistant head coach and defensive coordinator. He had been out of coaching since 2013, after he was fired by the Buccaneers following an 11-21 tenure in two seasons.
Schiano went 68-67 in 11 seasons as Rutgers' coach from 2001-11, guiding the Scarlet Knights to six winning campaigns and six bowl games in his last seven seasons there.
The Volunteers fired Butch Jones on Nov. 12, the day after they lost 50-17 at Missouri to fall to 0-6 in SEC play. He had a 34-27 record in five seasons at Tennessee. The Volunteers lost to Vanderbilt 42-24 to finish 4-8, 0-8 in the SEC, their first eight-loss season in school history and their first winless campaign in the SEC.
ESPN's Mark Schlabach and Chris Low and The Associated Press contributed to this report.