Byard and the Bears agreed Sunday on a two-year, $15 million contract, Ian Rapoport of NFL Network reports.
Philadelphia plans to release Byard in a move that frees up over $13 million in cap space, Adam Schefter of ESPN reports.
Byard finished the year with career highs in tackles (122) and solo tackles (80) in 16 regular-season games.
Byard recorded 13 tackles (nine solo) in Sunday's 37-34 overtime win versus Buffalo.
Byard is active for Sunday's Week 8 contest against Washington.
Tennessee traded Byard to the Eagles on Monday for Terrell Edmunds and 2024 fifth- and sixth-round draft picks, Dianna Russini of The Athletic reports.
Byard recorded 10 tackles (eight solo) in Sunday's 24-16 loss to Baltimore.
Byard and the Titans agreed on a reworked contract Friday, Terry McCormick of TitanInsider.com reports.
Byard finished the 2022 season with 108 tackles, six passes defended and four interceptions across 17 games.
Byard recorded eight defensive tackles (five solo) and one tackle on special teams Saturday in a loss to the Texans.