"Come back to the Jets," Johnson said at the NFL owners meetings, sounding almost like a college recruiter.
Team officials have showered Fitzpatrick with praise in recent weeks, but the two sides remain far apart on a new contract. The Jets are offering about $7 million per year, sources said, while Fitzpatrick is thought to be seeking twice that amount.
Johnson said he is optimistic a deal can be reached.
"I kind of hope so because he had such a good year last year," he said of Fitzpatrick, who set the franchise record with 31 touchdown passes last season. "I mean, it seems like an ideal [situation].
"I like him a lot as a person," Johnson continued. "He's very engaging. He's great with kids, great with our fans, great with our players. They really respect him. He's two guys. He's the Harvard guy and then he's the normal guy. They all make fun of him for being a Harvard guy, but he's a normal guy, too."
Johnson said he's not surprised negotiations have lasted this long, adding that it's part of the free-agency process. He said there's no sense of urgency, hinting that Fitzpatrick could afford to miss the offseason program because he has such a grasp of Chan Gailey's offense.
Ideally, the Jets don't want the standoff to last that long. At some point, their fallback options could sign elsewhere.
Robert Griffin III, who visited the Jets last week, also has drawn interest from the quarterback-needy Cleveland Browns. Johnson said the Jets were "very impressed" with RG III as a person, although it doesn't appear that the Jets have strong interest in the former Redskins quarterback.
Johnson declined to discuss the Jets' contingency plan, suggesting they won't act out of desperation if Fitzpatrick leaves. He noted that not every championship team needs an elite quarterback, mentioning the Denver Broncos, whose strong defense allowed them to overcome a diminished Peyton Manning.
"How did they win last year? A good D, baby," Johnson said. "How 'bout that Von Miller? How'd you like a guy like that? There's only one Von Miller."
The Jets' search for a franchise quarterback has lasted for decades. Johnson lamented the failures of his previous scouting department.
"We could've drafted Russell Wilson in the second round [in 2012]," he said. "We've had shots at players, but we didn't take those. Hopefully, we're getting better at evaluating talent, particularly at quarterback."