Watson was suspended for the first four games of the season due to a violation of the NFL's performance-enhancing drug policy. New England released him last week.
LaCosse is expected to miss at least a few weeks after suffering a sprained MCL during Thursday's win over the New York Giants, reports NFL Network's Michael Giardi.
Watson, a former first-round pick of the Patriots, spent the 2018 season with the New Orleans Saints. The veteran caught 35 passes for 400 yards and two touchdowns in 16 appearances.