The Warriors made the moves Wednesday, giving themselves the right to match any offer sheet signed by either player.
Morrow's offer is worth $1.029 million for next season, while Watson's offer is for $1.25 million.
Morrow averaged 13 points and 3.8 rebounds in 69 games with Golden State last season, making 37 starts. He was fifth in the NBA in 3-point percentage, making 45.6 percent.
Watson averaged 10.3 points and 2.8 assists in 65 games, including 15 starts.