The Texans originally signed Williams in 2014 as an undrafted rookie out of North Dakota State, but he failed to make the roster.
He spent the past four seasons with the Jets, who released him this week.
Williams, 26, has nine career interceptions and 101 tackles.
Banks originally was a second-round pick of the Buccaneers in 2013. He also has played for the Bears and Lions.
Banks played in three games for the Texans this season.
The Texans also released tackle Jah Reid.