Wilshere has struggled to hold down a place in the Gunners' starting lineup since returning from a loan at Bournemouth last season after fracturing his leg.
The England international is now fully fit and has been linked with a move away from the Emirates Stadium ahead of the expiry of his contract and the World Cup finals next summer.
One of the supposedly interested clubs is Palace, who are managed by a known admirer of Wilshere's in ex-Three Lions boss Hodgson.
But the 70-year-old has denied speculation linking Wilshere with Selhurst Park and claimed his representative is feeding baseless information to the media.
"Who has linked him? That's what you need to find out, because it didn't come from us," Hodgson told reporters.
"I will tell you who, his agent - because that's where newspapers get their stories from.
"We have never mentioned his name. I work with [Palace sporting director] Dougie Freedman and we talk about potential players, and Jack Wilshere's name has never been mentioned in this room.
"I like him and I think he is a very good player. There may come a time at Crystal Palace where Jack Wilshere would be just the player we are looking for, but I can't tell you that is the case at the moment."