If Osemele doesn't return to practice Saturday after sitting out nearly four weeks, he will be subject to a fine or a suspension for conduct detrimental to the team, a source said.
Osemele was cleared to practice by the team doctor and an independent doctor, the source said, adding that Osemele has a preexisting condition that can be addressed after the season.
Osemele, who usually avoids the media period in the locker room, made himself available Friday and told reporters that he needs surgery to repair his shoulder.
Osemele hasn't played or practiced since Sept. 22, when the Jets lost to the New England Patriots. The Jets (1-4) host the Patriots (6-0) on Monday night.
"It's been killing me," Osemele told reporters. "I'm just trying to get this done. I've done everything I can. I've been at work every day, waking up at 5 in the morning, doing all the rehab and the treatments and stuff like that. I'm like the last dude out of here at night. I'm doing everything I can.
"I'm working with my agent. We're communicating with the team. There's just not communication between the team and my doctor and my agent. It's just been butting heads for whatever reason. Hopefully, it gets resolved soon."
Another source refuted the Jets' claim, saying Osemele wasn't cleared by two doctors and that surgery was recommended. The source also said Osemele had to take the painkiller Toradol, prescribed by the Jets' doctor, so he could play in the first three games.
Osemele's agent, Andrew Kessler, said in a statement to the Associated Press that the Jets are not handling his client's situation correctly.
"Kelechi has a torn labrum, there is no debate about this as every doctor who has evaluated him agrees about that,'' Kessler said. "He has been advised by the second-opinion doctor that the timing of the need for surgery is based on his symptoms. His symptoms currently dictate that he needs surgery. For the team to question the integrity of how he has told them he is physically feeling is disappointing, to say the least.
"It is hardly putting a priority on player safety.''
Osemele said he hurt his shoulder Aug. 5 in practice, then aggravated it against New England.
He wasn't playing well at the time of the injury, and many believe he was on the verge of losing his starting job to Alex Lewis. After the bye in Week 4, with Osemele sitting out, the Jets turned to Lewis, who had started the past two games.
Two weeks ago, Osemele hurt his shoulder in a full-pads practice, coach Adam Gase said at the time. He said Osemele was "pretty banged up," noting he also was dealing with a knee injury.
The Jets acquired Osemele, 30, in a March trade with the Oakland Raiders. They sent a fifth-round pick to the Raiders for Osemele and a sixth-round pick.
Osemele is making $10.2 million this season. He's signed for 2020 at a non-guaranteed $11.7 million.