NFL Media first reported the news. The deal isn't yet official.
Trevathan, who turns 30 later in March, was set to become a free agent next week after originally signing with the Bears on a four-year deal in March 2016. He has played in 45 games over four seasons in Chicago, recording 327 total tackles. Trevathan is considered one of the most important figures in the Bears' locker room.
"He's such a great leader," Bears general manager Ryan Pace said in February. "He's such a good player.
"He's an important part of what we're doing."
The Bears entered this offseason with two free-agent inside linebackers in Trevathan and 26-year-old Nick Kwiatkoski. Chicago viewed the younger Kwiatkoski as a top offseason priority, a source said in February, but Pace and his brass understood his price could exceed their capabilities.
Kwiatkoski could command close to $30 million on the open market. With the Bears committing to Trevathan, Kwiatkoski will be coveted by other suitors.
The Bears had just shy of $25 million in salary cap space entering Thursday, according to Spotrac. While contract figures haven't surfaced yet for Trevathan, a team-friendly deal with him would position the Bears to be more aggressive in pursuing other positions, such as quarterback.
The NFL's free agency period begins next Wednesday.