The Gunners failed with an opening £40 million offer for the Ivory Coast international on Monday but have persisted with their interest.
Sky Sports News believes that Palace value the 26-year-old at £80 million and are resistant to sell after allowing Aaron Wan-Bissaka to join Manchester United in a £50 million deal this summer.
Reporter Kaveh Solhekol has claimed that an improved bid is yet to be made but the Gunners intend to submit a new offer.
Arsenal are said to have the means to push their bid as far as £70 million, but would walk away from a deal costing them more in total.
Zaha scored 10 Premier League goals in 34 Premier League appearances for the Eagles last season.
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Quiz! Can you name the 60 most expensive player transfers ever?
Transfers 11 players who re-joined their former clubs for (much) more money
