Skrine, a two-year starter for the Browns, will receive a four-year, $25 million contract, including $13 million in guarantees, the source said.
Later Tuesday, the Jets made a big splash by signing free agent Darrelle Revis. Going into free agency, the Jets had no healthy starting corners. Former No. 1 pick Dee Milliner is recovering from Achilles' tendon surgery and likely won't be ready until training camp.
Skrine, 25, earned a lot of respect while with the Cleveland Browns because of the way he competes and the fact that he improved every season.
Though not a No. 1 corner, Skrine is a hard-working, dependable second or third corner.
Skrine broke up 49 passes in four season seasons, but had just four interceptions in 2014 while ranking second overall in penalties with 15 (11 accepted). That ranking was partly due to the increased emphasis on illegal contact, as four of the six most-penalized players in the league were cornerbacks.
Pro Football Focus reported teams completed 61.5 percent of the passes thrown Skrine's way, and that he was responsible for eight touchdowns.
ESPN.com Browns reporter Pat McManamon contributed to this report.