Nugent, 35, is a former second-round pick of the Jets back in 2005. He has spent the past six seasons with the Bengals but has really struggled this year.
Nugent was in the final year of his two-year, $3.5 million contract and set to make a base salary of $1.2 million for the 2016 season when the Bengals cut him loose in December of last year.
The Giants signed Nugent to a contract a month ago before cutting him loose this past weekend.
In 2016, Nugent has converted 23 of 29 field goal attempts (79.3 percent) and 23 of 29 extra point tries to go along with 28 touchbacks.