Saleh said he feels "sick" for Becton, who has a chip fracture to his right knee. He also defended Becton against the "shade" he has received on social media.
"His story's not over. He's got full support of this organization," Saleh said. "And if you're a fan that wants to support him you're more than welcome, otherwise just keep it moving."
Becton is scheduled to get a second opinion on Wednesday. The Jets were initially optimistic about Saleh's injury but that changed Monday night after results of his MRI were known.
"Everything happens for a reason...I know God and my Grandma got me up there I JUST KNOW IT! TRUST!#IMSTILLHIM #4GK," Becton posted on Instagram.
The 2020 first-round draft pick injured the same knee that required season-ending surgery in September, but this is a different injury, Saleh said.
Late Monday, the Jets signed well-traveled veteran Caleb Benenoch (23 career starts) to a one-year contract. They remain interested in free agent tackle Duane Brown, a five-time Pro Bowl selection who visited with them over the weekend.
Becton has started only 14 out of 33 games in two seasons, including six games he didn't finish because of various injuries. His weight became a major storyline last season, when he reached about 400 pounds as he recovered from surgery.
He was overweight at the June minicamp, which frustrated the organization, but he trained at the facility before camp and reported in acceptable shape. Saleh had praised Becton for powering through the first nine practices as he worked his way back into football shape.
ESPN's Rich Cimini contributed to this report.