The Dutchman is one of a number of left-backs the Gunners are looking at as cover for first choice Kieran Tierney, though any deal is likely to take place this summer.
Van Aanholt who has become frustrated by a lack of game time at Palace having been shuffled in and out of the side with Tyrick Mitchell, another Arsenal are monitoring, impressing.
Asked if he expected any move from Arsenal for Van Aanholt, Hodgson said: “Not that I know of. Is Patrick Van Aanholt a good enough player to play for Arsenal? Yes he is but I think he is good enough to play for a lot of clubs, not least our own.
“This is where he is contracted and this is where I am expecting him to stay.”