The former Houston Astros skipper was identified as the clear front-runner after interviewing for the job on Tuesday, according to Morosi and Jason Beck of MLB.com.
MLB suspended Hinch for the 2020 season after its investigation into the Astros' sign-stealing scheme. Houston subsequently fired Hinch and replaced him with Dusty Baker.
If hired, the Tigers would be Hinch's third stop as manager after the Astros and Arizona Diamondbacks. In five years with Hinch at the helm, Houston went 481-329 and won the 2017 World Series, though the controversy stemming from the sign-stealing scandal has marred that season.
The 46-year-old former catcher spent one campaign of his playing career with the Tigers, hitting .203 with three home runs over 27 games in 2003.
Hinch had also been mentioned as a possible candidate for the Chicago White Sox job, but he didn't receive an interview before Tony La Russa was hired on Thursday, Bob Nightengale of USA Today Sports reports.
Former Tigers manager Ron Gardenhire abruptly retired in late September, citing health reasons.