Joel Sherman of the New York Post reported Shelton is set to join a list of candidates that includes former All-Star outfielder Carlos Beltran, former New York Yankees manager Joe Girardi and Arizona Diamondbacks vice president of player development Mike Bell.
Shelton spent time in the Yankees' minor league system as both a player and a manager after a standout collegiate career at Southern Illinois University.
The 49-year-old Illinois native started his major league coaching career as the Cleveland Indians' hitting coach in 2005. After five years with the Tribe, he moved to the Tampa Bay Rays in the same role for seven seasons starting in 2010. He also spent one year on the Toronto Blue Jays' staff in 2017.
Shelton spent the past two seasons as the Twins' bench coach.
In March, Twins chief baseball officer Derek Falvey and relief pitcher Trevor May explained his impact to La Velle E. Neal III of the Star Tribune.
"The ability to help fill for Rocco [Baldelli] in certain areas so Rocco can be focused more on big picture, the environment and culture," Falvey said. "And Derek can focus on execution daily. That's been a perfect partnership."
May added: "We love Sheltie. What he's doing now, he's really good at. He's a guy that, if something doesn't have value, we're not going to do it just because we have done it in the past. And it fits in well with a lot of analytic stuff."
The Mets haven't provided a timetable for a decision on the replacement for Mickey Callaway, who was fired following the 2019 regular season.