Hill, a 2018 third-round pick out of NC State, did not play in Sunday night's preseason finale against the New England Patriots with the trade in the works. The Giants had been scouring the league for offensive-line help in recent weeks.
Price, a 2018 first-round pick, provides the Giants with some much-needed depth on the interior of their offensive line. He has played center and guard in the NFL.
Price has played in 42 games with 19 starts since entering the NFL.
Hill, 26, has 7.5 sacks and 116 tackles in 48 games (17 starts). He started 12 games his rookie season and had 5.5 sacks, then settled into a rotation on the defensive line the past two years with the Giants.
Price, whom the Bengals selected with the 21st pick in 2018, suffered a foot injury in Week 2 of his rookie year that forced him to miss the next six games. He returned for the final eight games and reclaimed his role as the team's starting center.
However, in 2019 and under a new coaching staff, Price lost the starting job to Trey Hopkins. Hopkins was given a contract extension at the end of the 2019 season.
ESPN's Ben Baby contributed to this report.