The Bengals would like to hire Taylor, 35, after Los Angeles' season ends, sources said. Because the Rams are still in playoffs, they can't negotiate a contract.
Taylor was the offensive coordinator for the University of Cincinnati in 2016, and sources said he would like to return to the city.
Taylor would replace Marvin Lewis, who parted ways with the Bengals after the season, ending a 16-year tenure.
A former star quarterback at Nebraska, Taylor began his NFL coaching career with the Miami Dolphins as a quarterbacks coach in 2012, helping to develop Ryan Tannehill, and took over offensive-coordinator duties during the 2015 season.
After spending a year with the Bearcats, Taylor returned to the NFL with the Rams under Sean McVay, for whom he served as assistant wide-receivers coach in 2017 before shifting to quarterbacks this season.
With Taylor overseeing Jared Goff, the Rams went 13-3 and ranked fifth in the NFL this season with 282 passing yards per game.