"I'm real excited about this, excited about becoming a Texan," Joseph told the newspaper. "This (free agency) has been a grind, and I'm glad to get it over and get back to work.
"The Texans have a great offense and a lot of talent on defense. I think I can help the defense. I've got experience, and I'm very competitive, and I'm ready to go to work."
A person with knowledge of Joseph's deal told The Associated Press it was for five years and worth $48.75 million, with $23.5 million guaranteed.
But Joseph, 27, is three years younger than Asomugha, and he intercepted nine passes in his last two seasons in Cincinnati. The Bengals drafted Joseph out of South Carolina with the 24th overall pick in 2006.
The revamping of Houston's porous secondary didn't stop there. The team also agreed to terms with safety Danieal Manning on a four-year, $20 million deal, with $9 million guaranteed, the Chronicle reported Thursday. Manning will receive $11.5 million over the first two years of the deal.
The Texans open training camp Sunday and will start work on shoring up the NFL's worst pass defense from 2010.
Houston allowed an average of 268 passing yards per game and 33 touchdown passes overall. Defensive coordinator Frank Bush was fired, and the team brought in Wade Phillips to install a 3-4 alignment.
But the Texans also needed an effective cornerback to shore up a secondary that gave up a league-high 18 receptions covering 40 yards or more. While Asomugha was considered the best free-agent cornerback available, Joseph has more interceptions (14) over the past five seasons than Asomugha (11).
The Associated Press contributed to this report.