Per Schwartz, Barkley is a "far superior pass catcher" who has "absolutely no character issues or concerns."
The Giants own the second overall pick in the 2018 NFL Draft, which will be held on April 26-28 at AT&T Stadium in Arlington, Texas.
"I thought all along this is the guy Dave [Gettleman, Giants general manager] wants," said a national NFL scout who knows Gettleman, per the newspaper. "And he might get him, if Cleveland is dumb enough not to take him at No. 1."
USC quarterback Sam Darnold is viewed as the top overall prospect in the draft by NFLDraftScout.com senior analyst Rob Rang in his most recent Big Board. Rang lists UCLA quarterback Josh Rosen as second, followed by Barkley, Notre Dame offensive guard Quenton Nelson and North Carolina State defensive end Bradley Chubb.
Barkley caused a mild stir when photographed in Giants gear while receiving a haircut earlier in the week.
As for matters on the field, Barkley could provide a jolt to a Giants backfield that has failed to produce a 1,000-yard rusher since Ahmad Bradshaw in 2012.
Barkley rushed 217 times for 1,271 yards and 18 touchdowns last season with the Nittany Lions, including a 92-yarder in a Fiesta Bowl victory over Washington. The consensus 2017 All-American also caught 54 passes for 632 yards and two scores, as well as returning two kickoffs for touchdowns.
Gettleman didn't rule out taking Barkley with the second overall pick when asked during the NFL Scouting Combine.
"The whole 'running back position is devalued' is a big myth," Gettleman said at the time. "You take the best players when you can. If a player is worthy of a second overall pick in any draft, you take him."
The Giants reportedly also had a favorable reaction to Chubb, NFL Network's Mike Garafolo reported.
Garafolo said the Giants were "extremely impressed" with Chubb following his visit to East Rutherford, N.J. Chubb was professional throughout the experience, introducing himself to everyone in the facility while walking around with a smile all afternoon, per Garafolo.
Gettleman complimented Chubb on his suit and tie as well as his overall attitude during the visit, according to Garafolo.
While NJ.com reported that the Giants would select Darnold with the second overall pick of the 2018 NFL Draft, should he be available, the team conceivably could use that selection on Chubb as a means of addressing the departure of defensive end Jason Pierre-Paul. The Giants traded Pierre-Paul to the Tampa Bay Buccaneers on March 22.
Chubb, who is the ACC Defensive Player of the Year and Ted Hendricks Award winner (national defensive end of the year), reportedly met with the team at a private workout at North Carolina State last month.
The 21-year-old Chubb recorded career highs in tackles (72), tackles for loss (23) and sacks (10) in 2017 after posting similar numbers the previous season (56-21-10).
The Giants acquired a third- and fourth-round pick in the 2018 NFL Draft for Pierre-Paul and a fourth-round selection.
Pierre-Paul, 29, recorded 68 tackles, eight sacks and two forced fumbles for the Giants last season. He played in all 16 games despite nursing hand, shoulder and knee injuries.