Fleener, 27, has been durable during his first four seasons with the Indianapolis Colts. He missed only four games -- all during his rookie season in 2012.
Watson, 35, had a career year for the Saints in 2015. He either set or matched career bests with 74 catches, 825 yards and six touchdowns after replacing Jimmy Graham as the top tight end.
Fleener is more of a pass-catcher than blocker. He's totaled 183 receptions for 2,154 yards and 17 touchdowns in his career. His best season was in 2014, when he had 774 yards receiving to go with eight touchdown receptions.
ESPN Colts reporter Mike Wells contributed to this report.