Skrine spent the last four seasons with the Cleveland Browns, who drafted him in the fifth round in 2011.
The 25-year-old reportedly heard from eight teams on Saturday, the first day of the NFL's free agent negotiating window.
Skrine started all 16 games for the Browns last season, recording 67 tackles, four interceptions and 18 passes defended.
The Jets finished 4-12 last season, missing the playoffs for the fourth straight year.
- Molly Geary