The Mundy signing came after the Giants lost free-agent safety Kenny Phillips, who signed a one-year deal with NFC East rival Philadelphia.
Mundy, 28, had a career-high 39 tackles in 16 games last season, starting three of those games for the Steelers. In his four seasons in Pittsburgh, he has 130 career tackles and one interception in five career starts. Mundy could be the team's third safety in Perry Fewell's three-safety look with Antrel Rolle and Stevie Brown, a restricted free agent who received a second-round tender.
Brown emerged last year after Phillips injured the medial collateral ligament in his right knee. Brown had eight interceptions and likely will start alongside Rolle this year. Brown's second-round RFA tender gives the Giants the right of first refusal if a team offers Brown a contract. The Giants can match and, if they opt not to, they will receive a second-round pick in return.
The Giants need Brown and Mundy after losing Phillips, the team's 2008 first-round pick. The team did officially announce on Thursday the return of its 2007 first-round pick, cornerback Aaron Ross. Ross agreed to a one-year deal on Tuesday.
Rivers, who played in 11 games in his first season with the Giants last season after coming in a trade from Cincinnati, will return for a one-year deal. The move was reported earlier by The (Newark) Star-Ledger.
Rivers, 26, could step in immediately for Michael Boley, who was released by the Giants in a salary-related move earlier this offseason. Rivers started in six games and had 44 tackles but was slowed by a variety of injuries.
When healthy, Rivers showed glimpses of being able to fill Boley's shoes in the Giants' defense. The Giants have a need at linebacker. Besides Boley's departure, starting middle linebacker Chase Blackburn is a free agent. The team could bring Blackburn back for a cheap short-term deal, but Blackburn is exploring his options on the market.
The team currently has Rivers, Jacquian Williams, Mark Herzlich and Spencer Paysinger at linebacker. The Giants had Minnesota free-agent linebacker Jasper Brinkley in for a visit Wednesday. They have expressed some interest in Cincinnati free-agent linebacker Rey Maualuga, a source told ESPN.
The Giants acquired Rivers from Cincinnati for a fifth-round pick in last year's draft. Originally picked ninth overall out of USC in the 2008 draft by the Bengals, Rivers has 155 tackles, two sacks and two interceptions in 46 games played.