The deal includes a $2 million signing bonus and $3 million in guaranteed money, according to the source.
Jenkins joined the Giants less than two weeks after being cut by the rival Eagles, who parted ways with the nine-year veteran following two sub-par seasons in Philadelphia.
Jenkins was part of the Eagles' free-agent spending spree in 2011 when he left the Green Bay Packers to join Philadelphia on a five-year, $25 million deal.
The Giants, who missed the playoffs in 2012 after winning the Super Bowl the previous season, finished 31st in defense last season.
The Giants needed to replace defensive tackle Chris Canty, who was released last month when the team was slashing salary. Jenkins is now slated to replace Canty in the starting lineup alongside Linval Joseph. The Giants also have Shaun Rogers, Marvin Austin, Markus Kuhn and Bobby Skinner at defensive tackle.
The 32-year-old Jenkins started all 32 games after signing with the Eagles and had 9.5 sacks in his two seasons for Philadelphia, including four last season. Jenkins, who spent his first seven NFL seasons in Green Bay, has 38.5 sacks in 125 career games.
The Giants need Jenkins to help plug the middle and stop the run. New York struggled defensively against the run last season, finishing 25th in the league while allowing an average of 129.1 rushing yards per game.
ESPN NFL Insider Adam Schefter, ESPNNewYork.com's Ohm Youngmisuk and The Associated Press contributed to this report.