Eyeing Zaha as part of a summer reshuffle that is yet to get off the ground, Arsenal were thought to have offered £40m for the Ivory Coast international earlier this month.
However, Palace are rumoured to value Zaha at around double that in the region of £80m and Hodgson has stated the Gunners simply haven't matched that figure.
"I don't think Arsenal have actually made a bid that has come anywhere near our valuation," Hodgson explained to Sky Sports following a 6-2 pre-season loss to Barnet.
"I'm sure the player realises that if someone's going to take him away from us, he'll expect clubs to pay the market value. Until someone does that, there's not much to discuss regarding Wilf."
Some of the gossip surrounding Everton alleges that a €28m (£25.3m) that could rise to €40m (£36.2m) has been agreed with Gremio for the 23-year-old, with club doctors even supposedly travelling to Brazil to conduct a medical.
Whatever Arsenal do in the transfer market this summer, Josh Kroenke - son of unpopular owner Stan, has warned that the club will not be in a position to compete for trophies in 2019/20.
"There are no shortcuts," the American said in a much publicised response to a supporters group letter criticising the club's ownership.
"As much as we would love to say that we'll be competing for the top trophies tomorrow, we know that's not the case and that we have to get to work on that. That's a daily approach to the little things from the top down.
"It's no secret that we have a Champions League wage bill on a Europa League budget right now. That's a fact. And one that we're figuring out how to face internally at the moment.
"That doesn't mean that my father or I have ever said anything about any restrictions on spending. It's not who you buy, it's how you buy."