Cole, a team captain whose 2019 season was cut short by a torn ACL in November, was tendered a qualifying offer of $2.144 million.
Cole has 61 tackles in 31 career games for the Texans. He was signed as an undrafted free agent out of Missouri State in 2017. In his rookie season, Cole became the third linebacker in team history with multiple interceptions in a season.
Cole had 16 tackles in 11 games last season.
General manager and head coach Bill O'Brien talks about Cole as one of the leaders in the Texans' locker room.
Cole is also a valuable member of the Texans' special teams unit.
O'Brien in November described him as "one of the better players in the league at his role."
Adams played in just five games last season. He recorded 13 tackles for the Texans.
O'Brien spoke highly of Adams after he forced a fumble on Tampa Bay Buccaneers running back Peyton Barber in Week 16.
Douglas DeFelice-USA TODAY Sports "When we've asked (Adams) to step up his role – sometimes he's been inactive and then other times we've asked him to come in and play on special teams," O'Brien said. "(In 2018), he blocked a punt, this year he comes in, he plays well on special teams then last week he causes a fumble. The guy's just, he's really one of those guys, in my opinion, that really exemplifies what pro football is all about. It's about the fabric guys and the role players doing their jobs at the highest level when you ask them to do that. That's what Tyrell does."