Scouts were spotted watching the 17-year-old during Saturday's Peninsula League clash with Okehampton Argyle, in which he was a second-half substitute.
But Appledore manager Geoff Copp is not prepared to lose the forward without a fight.
He said: "He is a great prospect and we will do our best to keep hold of him.
"I think everybody can see that it is definitely in him, but he still has a lot of growing and developing to do before he becomes the player that he could be.
"If he gets the chance at a bigger club as a YTS trainee then he would learn his proper trade with quality coaches and better people around him so he would only improve as a player.
"I will just ask the club to do their best to keep my players." Copp, however, insists that he will not stand in Downing's way if any move develops.
He said: "It will come down to money and how can I be hypocritical and tell him not to go if money is being chucked at him?"