The Gunners would appear to have cooled their interest for now, with Goal able to confirm that those at Emirates Stadium are close to wrapping up an €80 million (£72m/$89m) deal for Lille forward Nicolas Pepe.
Hodgson is in the dark when it comes to those reports, with the Eagles manager unsure what is happening with a player yet to return to his squad from Africa Cup of Nations duty.
He told Sky Sports when asked for an update on Zaha’s situation: “He’s dealing with the chairman [Steve Parish].
“The chairman’s his link, they know each other very well and get on very well. Most of what’s going on with Wilf is going on between him and the chairman and I can’t enlighten you at all.
“What I know, you know. I know it through reading your websites and newspapers that this or that is happening, but I can’t enlighten you at all.”
Quizzed further on whether he was aware of a bid being lodged by Everton, Hodgson said: “I don’t know.”
The Eagles boss has previously expressed a desire to keep Zaha on his books, with Palace having already sanctioned the £50m ($62m) sale of Aaron Wan-Bissaka to Manchester United this summer.
He has said: “He [Zaha] has not spoken to me. I don't know if he's spoken to the club [asking to leave].
“As far as I'm concerned, he is coming back, he is our player and I've been given no information that the club is interested in selling him at the moment.”
Hodgson added: “I'm just expecting him back, looking forward to working with him and seeing some of the things he can do for us.
Palace faced Bristol City on Saturday, with Hodgson’s side running out 5-0 winners against their Championship opponents at Ashton Gate.