Geno Smith hasn’t played a down in the NFL yet, and already questions about his character and ability to lead a professional team have cropped up.

On Wednesday, Yahoo! reported that Smith — while visiting prospective teams on pre-draft meetings — sat by himself playing with his cell phone, sending text messages and scrolling through Twitter instead of chatting with team execs and coaches.

“All these other players who were in there were talking to the coaches, trying to get to know people and he was over there by himself,” one source said to Yahoo! “That’s not what you want out of your quarterback.”

That report was released a day after Smith fired his agents from Select Sports Group. The Daily News reported Tuesday that Smith felt that he would have and should have been selected with the first pick of the draft. Instead, Smith fell all the way to the Jets in the second round as the 39th overall selection, quickly firing his agents after he was forced to wait through the first round at Radio City Music Hall last Thursday without hearing his name called.

Smith considered leaving New York, but eventually changed his mind and decided to stay for the second round.

Smith, who will compete with embattled starter Mark Sanchez in training camp, was under attack even before he became a Jet, as Pro Football Weekly gave his personality a powerfully negative review. “Not a student of the game. Nonchalant field presence — does not command respect from teammates and cannot inspire,” that report read. “Mild practice demeanor — no urgency. Not committed or focused — marginal work ethic.”

Smith has been professional, if not boring, in his limited exposure to the New York media thus far. But the report Wednesday perhaps shed some light on why the quarterback, who was considered a solid bet for one of the top 10 picks in this year’s draft, fell all the way to the second round.

“His biggest problem is that he doesn’t know what he doesn’t know,” a league said executive in the report. “I’m not sure he knows how to take instruction because he pretty much wouldn’t listen or talk to our coaches . . . He’s talented. He can sling it, he can fit it into tight spots, he can do a lot of things and I think he wants to be good.

But you can’t tell him anything right now. He’s tuned out because he thinks he’s got it all down.”