The deal is worth up to $6 million, a source told ESPN's Adam Schefter.
"Just excited about the opportunity," McCourty said in a text message to ESPN's Paul Kuharsky. "Got a chance to be around the staff over the past day and half and a chance to sit and talk with some of the players. Very clear to see the excitement going on in the building. Just looking forward to helping this team in any way I can."
McCourty played for the Tennessee Titans the past eight seasons, but became a free agent after he was released on April 18 when he and the team could not agree on a reduced contract for 2017.
He started every game he played in his last six seasons for the Titans.
McCourty, the twin brother of Devin McCourty of the Patriots, has 13 interceptions and 72 passes defenses in his NFL career.