Brown, 27, was a fourth-round pick by the Giants in 2009, but he didn't blossom as a player until 2012, when he served as the team's goal-line back and scored eight rushing touchdowns before breaking his leg with five games left in the season.
He broke the same leg in the same spot in the final preseason game of 2013 and missed the first eight games of the year on injured reserve. He had been slated to open the season as the "co-starter" at running back with 2012 first-round pick David Wilson.
When Brown returned, Wilson was on IR. Brown assumed the role of starting running back and rushed for more than 100 yards in two of his first three games. His production tailed off down the stretch, though, and he fumbled three times in the final four games of the season.
KRIV-TV and the Houston Chronicle had earlier reported Brown had agreed to sign with the Texans.
ESPN.com Giants reporter Dan Graziano contributed to this report.