Barkley was selected by the Philadelphia Eagles, who traded up with Jacksonville to get the first pick in round No. 4, and Nassib was nabbed by the New York Giants.
Both had been projected as going higher in the draft, which leaned more heavily on offensive linemen than on quarterbacks. E.J. Emanuel of Florida State was the only quarterback drafted in the first round.
Barkley threw for 39 touchdowns at USC. The Trojans were familiar foes of Oregon Coach Chip Kelly, who took the Eagles head coaching job in the off-season.
Two other quarterbacks were taken in the fourth round. Landry Jones of Oklahoma went to the Pittsburgh Steelers, and Wilson Tyler of Arkansas was picked by the Oakland Raiders.
The draft continued Saturday afternoon.