Richardson, acquired from the New York Jets just prior to the 2017 season, is coveted by the Seahawks, NFL.com reported, but not at the $14 million salary he would receive with the franchise tag.
Carroll said during his season-ending news conference that it would be "huge" to have Richardson return to the team.
The 6-foot-3, 294-pound Richardson started 15 games for the Seahawks last season, registering 44 tackles and a career-low one sack. He spent the previous four seasons with the Jets, earning Pro Bowl honors in 2014.
Seattle paid a hefty price to bring Richardson to the Pacific Northwest, sending wide receiver Jermaine Kearse and a second-round draft pick to New York.
The deadline for teams to apply the franchise tag on a player is March 6.