Thurmond inked a one-year contract worth $3.5 million, a league source told ESPN NFL Insider Adam Schefter.
Thurmond, 26, could fit as a starter with the Giants after starting three games for the Super Bowl champions last season, finishing with 24 tackles, one interception and one sack. The 5-foot-11, 190-pounder also served a four-game suspension for violating the league's substance abuse policy.
Safety/kick returner Quintin Demps, who made six starts for the Kansas City Chiefs last season, said via Twitter on Saturday that he's also heading to the Giants. The Giants confirmed Sunday that they've signed Demps.
Demps, returning to the NFC East after starting his career with the Philadelphia Eagles, registered a career-high four interceptions last season.