Sports Illustrated first reported on talks between Rutgers and Ash.
A news conference to introduce Ash is expected Monday, a source told The Associated Press.
Ash has coached the Buckeyes' defense since 2014. This season, No. 6 Ohio State (11-1, 7-1 Big Ten) ranks second in the nation in scoring defense (14 PPG) and 10th in total defense (303.5 YPG).
Ash would replace Kyle Flood, who was fired after a 4-8 campaign, as Rutgers seeks a fresh start after a season filled with off-field troubles.
Flood was suspended for three games this season for emailing a professor and later scheduling an in-person meeting to discuss the academic standing of former Scarlet Knights defensive back Nadir Barnwell.
In addition, seven Rutgers players were arrested this season on charges varying from home invasion to assault.
Athletic director Julie Hermann was also fired the day after the regular-season finale. Patrick Hobbs was hired to replace her.