The 47-year-old played only 10 times for the Third Division side, including his landmark 1,000th, after joining them from West Ham last November.
Orient's manager, Tommy Taylor, said: "Peter is magnificent for his age and if he could kick the ball I would keep him. It sounds silly but you need a goalkeeper to kick it into your opponent's penalty box to make chances in this division."
Orient's chief scout, Steve Shorey, added: "I wouldn't like to say anything against a player who has served the game so well... other clubs might be interested in him."
He pointed out that Shilton, who is to play several games in Hong Kong, was Orient's first choice until he was injured and the club signed Paul Hyde, who has taken over the No 1 jersey.