Bowles also mentioned the possibility of selecting a quarterback with the No. 6 overall pick, saying "there is a scenario" where that could occur.
Some assumed the 37-year-old McCown, with 14 years of experience, would be given the inside track after signing a one-year, $6 million contract (fully guaranteed) last week. But the organization wants to give Hackenberg, a second-round pick in 2016, a legitimate shot at the job.
"There will be heavy competition for the job," Bowles said. "Nobody has been promised the starting quarterback job. They'll all get a chance to play and we'll make that decision, going forward, when training camp starts as we see production from certain people."
The immediate plan is to have them "alternate reps" through the offseason, Bowles said. The organized team activities (OTA practices) begin May 23 and continue through June 15 with the conclusion of minicamp. At that point, they will re-evaluate.
Petty's status is up in the air, as he's still recovering from labrum surgery on his non-throwing shoulder. Bowles said he's not sure when Petty will be able to resume throwing.
The Jets are rebuilding at quarterback because their most experienced signal callers from last season are gone. Ryan Fitzpatrick is a free agent (he won't be re-signed) and Geno Smith signed a free-agent contract with the New York Giants.
In his first two seasons, Bowles declared a No. 1 quarterback at the start of training camp because he wanted the presumptive starter to get the bulk of the practice time to prepare for the season. He'd like to follow that plan, assuming one player separates from the field.
Essentially, Bowles is looking for a game manager.
"We just have to make sure we have a guy that doesn't turn the ball over and we have a guy that can manage the room, and make sure we help that guy and not put it solely on the quarterback," he said.
The organization is hoping Hackenberg can take a big leap in Year 2. He played poorly last preseason and struggled during in-season practices, although he never got much quality practice time because of a four-man depth chart.
"This year, he'll have a chance to play and he'll have some pep in his step," said Bowles, who promised to be more involved in the offense under new coordinator John Morton.
As the draft approaches, the Jets have been linked to Deshaun Watson and Mitchell Trubisky with the sixth pick.
"We'll see how it falls and we'll look at the pros and cons of it and we'll make that decision, but there is a scenario, yes," he said.
Asked which quarterback he prefers, Bowles said without hesitation, "Roger Staubach."