Soliai's deal is for five years and $33 million, with $14 million guaranteed and $11 million coming in the first year, sources told ESPN NFL Insider Adam Caplan.
Soliai, 30, was a fourth-round pick of the Miami Dolphins in 2007. He finished with 34 tackles and one sack last season.
He also was the Dolphins' best run-stopper. However, Miami ranked 26th in the league in run defense.
Jackson, 27, was the Kansas City Chiefs' first-round draft pick in 2009 and the third overall selection that year, though he never played to that lofty stature.
He was an immediate starter but never developed his pass-rush skills to the point where the Chiefs felt comfortable leaving him in the lineup in obvious passing situations. He had his best season in 2013. He was productive against the run and had a career-high four sacks.
Asamoah also comes from the Chiefs, who drafted him in the third round in 2010. He is coming off a season in which he played in a career-low 13 games.
The Falcons also released Pro Bowl safety Thomas Decoud on Tuesday.
Information from ESPN.com Dolphins reporter James Walker and Chiefs reporter Adam Teicher contributed to this report.