Sources told ESPN's Rachel Nichols that Ryan will accept the position. A formal announcement is expected Monday, with a news conference likely on Wednesday.
A Jets team spokesperson declined comment.
The Jets notified other candidates, including offensive coordinator Brian Schottenheimer, of their plans to hire Ryan, the sources said.
Ryan told Nichols late Sunday that he might find out about the Jets' job as early as Sunday night.
"We'll see. There were no pre-arrangements made or anything, though. I was completely focused on our team."
When asked if he would accept the Jets' job if offered, Ryan said: "Sure. Any coach who believes in himself wants the opportunity, and it's a great franchise."