Also Monday, defensive tackle Mike Patterson re-signed with the Giants. The team did not disclose terms of the contracts with the two players.
Bowman, who visited the Giants on March 24, had 48 tackles and three interceptions last season.
The 29-year-old spent six years with the Bears. His best season was 2009 when he had six picks and 66 tackles.
The Bears are believed to have expressed a desire to bring back Bowman.
Patterson spent last season with the Giants after eight years in Philadelphia. He played in all 16 games for the first time since 2009 and made 35 tackles (22 solo).
In nine years, Patterson has played in 131 regular-season games with 100 starts, and started seven postseason games.
ESPNChicago.com's Jeff Dickerson and The Associated Press contributed to this report.