Terms were not disclosed. It's a one-year, $6.5 million deal, a source told ESPN's Adam Schefter.
Melvin, 28, went from being projected as the Colts' third cornerback at the start of training camp last season to replacing Vontae Davis as the No. 1 cornerback.
He started 10 games before suffering a season-ending hand injury against Tennessee in Week 12.
Melvin's aggressive style allowed former coach Chuck Pagano to shift from having left-right defensive assignments at cornerback to having Melvin shadow the opposing team's best receiver.
Melvin, who has played at least 10 games in a season only twice in his five years in the NFL, had three interceptions and 13 passes defended last season.
ESPN's Mike Wells contributed to this report.