Joseph told the newspaper that his decision came down to the Texans and Oakland Raiders, but in the end it would have been tough to move his family after being in Houston for seven seasons. He said money wasn't a factor.
He will sign a two-year deal, ESPN's Adam Schefter reports.
In 2017, Joseph had 45 tackles and two interceptions, including a pick-six against the Cleveland Browns in Week 6. He said at the end of the season that he hopes to play at least two more years in the NFL.
"He's one of the leaders of the locker room, been that way since we got here," Texans head coach Bill O'Brien said last season. "He's asked to do a lot on our defense relative to covering top guys. He's been doing that since we arrived here, but he's a team guy."
Joseph, 33, was drafted in 2006 by the Cincinnati Bengals and he spent his first five seasons with them.
He has been selected to two Pro Bowls in 12 NFL seasons and has 28 interceptions, 172 passes defensed, 7 forced fumbles and 645 tackles in his career.
ESPN's Sarah Barshop contributed to this report.