The team also said it had cut linebacker Kareem Martin.
Ogletree, one of the league's highest-paid inside linebackers over the past two years at $10 million per season, became a liability in pass coverage and saw his playing time diminish.
Ogletree, who turns 29 in September, was one of general manager Dave Gettleman's first major acquisitions with the Giants. They thought Ogletree could be plugged into the middle of the defense and make a significant impact after being acquired from the Los Angeles Rams in 2018 for two draft picks.
He played in 13 games each of the past two seasons and had 80 tackles, a sack and an interception last year. Ogletree had 93 tackles, a sack and five interceptions in 2018.
The veteran was again slated to make $10 million in 2020 and was scheduled to be the fourth-highest-paid inside linebacker in the NFL, but none of the money was guaranteed. The Giants will take a dead money hit of $3.5 million by cutting Ogletree but will save $8.25 million against the salary cap.
Ogletree played the first five years of his career with the Rams, topping 100 tackles in three of his first four seasons. The second-team All-Pro in 2016 has career totals of 674 tackles, 7.5 sacks, 12 forced fumbles and 12 interceptions. He also has four defensive touchdowns.
The Giants signed Martin to a three-year, $21 million contract one day after acquiring Ogletree in March 2018. Martin spent his first four seasons in the league with the Arizona Cardinals.
Martin played in all 16 games in his first season in New York and had a career-high 48 tackles. But a knee injury in the 2019 season opener limited him to five games last season.
ESPN's Jordan Raanan contributed to this report.