Barkley was projected to be the fifth USC quarterback drafted in the first round, but every team passed on him multiple times in the first three rounds.
On Saturday, coach Chip Kelly said Barkley will be included in a "wide open" competition for the starting position.
"I'm an Eagle!! Time to fly baby," Barkley wrote on Twitter.
Nassib was the fifth quarterback drafted in 2013. He was chosen 110th overall, two picks before the Oakland Raiders snagged Wilson. Then at No. 115, the Steelers drafted Jones.
All three enter the NFL as backups in an era in which rookies more commonly earn starting spots.
The Steelers have been outspoken about their need to find a steady second option behind Ben Roethlisberger and were at Jones' Pro Day in Norman. They also met with the Sooners' four-year starter at the 2013 Scouting Combine.
Nassib, one of the top 50 players in the 2013 class according to NFLDraftScout.com, had a 26-to-10 TD-to-INT ratio as a senior. The product of Giants coach Tom Coughlin's alma mater is the first Syracuse quarterback drafted since Donovan McNabb in 1999. He shouldn't plan on a regular role immediately.
Eli Manning currently leads NFL quarterbacks with 135 consecutive regular-season games played.
Wilson, who replaced Ryan Mallett as Arkansas' starter in 2011, was hindered by a weaker supporting cast and unexpected coaching change just before the season when Bobby Petrino was fired. He steps into a less daunting competition for time with veteran Matt Flynn listed first on the Raiders' depth chart. Flynn has never been a fulltime starter and lost a QB competition to Russell Wilson with the Seattle Seahawks in training camp last summer.