Financial terms of the deal weren't immediately available. Idonije visited the team's facilities Tuesday at Halas Hall, where he took a physical.
A source told ESPNChicago.com's Jeff Dickerson that Bennett refused to take a pay cut.
Idonije spent the first nine years of his career with Chicago before joining the Detroit Lions in 2013. In one season with the Lions, Idonije participated in 15 games and contributed 11 tackles and a half sack.
Prior to joining the Lions, Idonije started his career in Chicago as a core special teams contributor, and later blossomed into a full-time starter.
When Idonije signed with the Lions, he was coming off a 2012 season with the Bears in which he finished second on the team with 7.5 sacks.
Despite the Bears having already signed defensive ends Lamarr Houston and Willie Young in free agency, the club wanted to add depth to a defensive line which sorely lacked it in 2014. Idonije provides scheme versatility because of his ability to line up outside at defensive end and inside at defensive tackle.
Many within the organization view Idonije as a potential positive influence in the locker room because of his experience with the franchise and work ethic.
Over Idonije's last four years in Chicago, he posted 23 sacks, and in 149 career games, the defensive end has logged 50 starts, 29 sacks and 203 career tackles.
Bennett took a pay cut in 2013 and lowered his salary by $1 million.
Bennett had a cap number of $1.35 million (after the reduction) last year, but was scheduled to count $2.45 million against the Bears' cap in 2014 and earn a total of $2.45 million (that included a $100,000 roster bonus).
Bennett finished 2013 with 32 receptions for 243 yards and four touchdowns, but had to miss the final game of the year against the Green Bay Packers to be with his ailing brother who died in the offseason.
Steltz, who made one start at safety last season, has been a key part of the Bears' special teams. McCray was primarily used there last season with Dallas.
Information from ESPNChicago.com's Jeff Dickerson and The Associated Press was used in this report.