Earlier this month, Sky Sports News reported Arsenal made a £40 million bid for Zaha, but Palace are not open to selling the 26-year-old and value him at £80 million.
Speaking after Palace's 6-2 pre-season defeat to Barnet on Tuesday, Hodgson confirmed the Gunners are a long way off tabling a bid that Palace would consider, per Adam Goodwin of Sky Sports:
"I don't think Arsenal have actually made a bid that has come anywhere near our valuation. 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."
Per Sam Dean of the Telegraph, the Gunners have a modest transfer budget this summer of around £45 million, plus any money they make from sales.
The close season has been frustrating for Arsenal:
But manager Unai Emery has made assurances they will sign "three, four players," which he said would be "very big players, very expensive," per Dean.
That will likely mean having to make some big money on outgoing stars, especially if they want to sign Zaha.
The Ivory Coast international returned 10 goals and five assists in 34 Premier League appearances for Palace last term.
At Arsenal, surrounded by the likes of Mesut Ozil, Alexandre Lacazette and Pierre-Emerick Aubameyang, Zaha could become even more dangerous in front of goal.
He would also provide Arsenal with extra pace down the right flank, where they are lacking options, and he could provide cover for the striker's role and the left flank as he is a versatile attacker.
One way Arsenal may be able to get the signing over the line is to include players on top of cash in the deal:
Having already lost Aaron Wan-Bissaka to Manchester United this summer, Palace will be loath to let another of their biggest stars leave the club.
But if they can bolster their squad at the same time as receiving a healthy fee for Zaha, they could well be tempted.