The news was first reported by NFL Network.
Barrett will face an uphill climb to crack the Saints' 53-man roster, although Brees' backup job is a wide-open competition after they let veteran Chase Daniel leave in free agency.
Head coach Sean Payton appears enamored with dual-threat quarterback Hill, who signed with New Orleans last September after being cut as an undrafted rookie by the Green Bay Packers. Hill excelled in special-teams kick coverage for the Saints.
New Orleans signed Savage, the former Houston Texans starter, to compete with Hill.
Brees is 39 years old, so the Saints will eventually need to find his successor. But they didn't draft one this year, and they don't have a first-round pick in 2019 after trading it to move up for defensive end Marcus Davenport in this year's draft.
Barrett had been invited to try out at the Indianapolis Colts' rookie minicamp this weekend before reportedly signing the deal with the Saints.
Barrett threw for 3,053 yards with 35 touchdowns and nine interceptions last season for the Buckeyes. He also rushed for 798 yards and 12 touchdowns.
ESPN's Mike Triplett contributed to this report.