The deal is worth $7.5 million, a source told ESPN's Dianna Russini.
It's a homecoming for Ryan, 29, who grew up in New Jersey and played for Rutgers in college.
The Giants were getting thin in the secondary, and Ryan has a history with Giants head coach Joe Judge, a former New England Patriots assistant.
Ryan will slide into the safety/nickel/corner role the Giants had slated for rookie Xavier McKinney, who fractured his left foot last week.
The news of Ryan's agreement was first reported by NFL Network.
The Giants had been monitoring Ryan's situation since the start of free agency. His salary was undoubtedly a factor, with Ryan originally looking for something in the range of $10 million.
That was too rich for the Giants, who entered Monday more than $24 million under the salary cap, per ESPN's roster management system. But the Giants waited, and Ryan become a serious option as the secondary became depleted: DeAndre Baker was charged with armed robbery; Sam Beal opted out of the season; and McKinney is out until late November with his injury.
The mutual interest began building in recent days, and the move was aided by Ryan's recent switch to New York-based agent Joel Segal.
Ryan was a jack-of-all-trades for the Tennessee Titans last season. He became only the third player since 2000 to finish with at least 4 sacks (4.5), 4 interceptions and 4 forced fumbles in a season.
Ryan posted on Instagram in May that his "chapter in Tennessee has come to a close" and later said on Devin and Jason McCourty's "Double Coverage" podcast that "Tennessee really never offered me a contract ... they never really tried to bring me back."
Ryan played primarily as a nickelback but was used frequently on blitzes and relied upon to be a force against the run. He started all 16 games last season for the first time in his career and set highs with 18 pass breakups and 113 tackles.
In the playoffs, his pick-six of former Patriots teammate Tom Brady sealed the Titans' upset victory in the wild-card round.
Since entering the league with the Patriots in 2013, Ryan has 494 tackles, 11 sacks, 8 forced fumbles and 17 interceptions, with one defensive score.
ESPN's Jordan Raanan and Turron Davenport contributed to this report.