Brissett, who started 15 games last season for the injured Andrew Luck, would require more than a first-round pick from any potential suitor, Colts owner Jim Irsay told the Indianapolis Star.
"All of a sudden, we have the best backup quarterback in football," Irsay said, per the Indianapolis Star. "I don't think we'd accept a (first-round pick in a trade) for him, we think he's that good."
While the Philadelphia Eagles likely can argue Irsay's point about backup quarterbacks -- their backup Nick Foles won Super Bowl MVP honors, after all -- Brissett showed flashes during the Colts' 4-12 season in 2017.
Brissett was acquired from the New England Patriots in September to serve as a backup to Scott Tolzien while Luck was rehabbing from shoulder surgery.
Instead, the 2016 third-round pick out of North Carolina State threw for 3,098 yards with 13 touchdowns against seven interceptions. Brissett, who also rushed 63 times for 260 yards and four scores, is under contract through the 2019 season.
Brissett, 25, started off his rookie season as a third-stringer but wound up appearing in three games, starting two, when New England quarterback Tom Brady was serving a four-game suspension and backup Jimmy Garoppolo was injured in Week 2.