The news was first reported by the New York Post.
Callaway will replace Terry Collins, who stepped down as manager to take a position in the team's front office, after the Mets fell to a 70-92 record this season.
Mets hitting coach Kevin Long, White Sox bench coach Joe McEwing, Mariners third-base coach Manny Acta and Astros bench coach Alex Cora also reportedly had interviewed for the job.
Callaway, 42, has built the Indians' pitching staff into one of the best in baseball since joining the team in 2013. Cleveland, who finished with an American League-leading 102 wins, led the majors in ERA (3.30), strikeouts (1,614), shutouts (19) and complete games (7), and finished third with an opponents' batting average of just .236.
Callaway also was influential in the Indians' decision to move Carrasco from the bullpen into the rotation in August 2014.
Prior to joining the Indians, Callaway also coached with the Rangers, Angels and Rays.