Starsport understands Unai Emery is eager to add the Eagles forward to his squad but cannot afford his £80m price tag.
Arsenal have already seen a £40m offer for Zaha, who is currently on international duty at the Africa Cup of Nations, knocked backed.
According to The Sun, a second deal is set to be proposed to Crystal Palace.
Arsenal will offer Calum Chambers, Mohamed Elneny and Carl Jenkinson in addition to the original £40m bid.
Emirates Stadium bosses value the trio of players at £40m, which they view as doubling their initial proposal.
Zaha, who grew up supporting Arsenal, is eager to make the move.
He is reluctant to submit a transfer request, however, and has four years left on the £130,000-a-week contract he signed last summer.
Crystal Palace are under no pressure to sell after banking £50m from the sale of Aaron Wan-Bissaka to Manchester United.
Selhurst Park chiefs agreed a 25 per cent sell-on clause on any future profit when the re-signed Zaha from the Red Devils in 2015.
During negotiations over the Wan-Bissaka deal, they failed in an attempt to have that clause removed.
And The Sun claim Arsenal bosses will highlight to Crystal Palace that accepting the players-plus-cash proposal will see them having to hand over less money to United.
Zaha is in action for Ivory Coast against Algeria in the Africa Cup of Nations quarter-finals this afternoon.
When asked about his future earlier this month, the 26-year-old said: “The only thing that’s on my mind right now is trying to win this cup.”
Earlier this month, Zaha’s brother Judicael told Sky Sports: “Wilfried will always hold Crystal Palace and their fans in the highest regard and all the support they have given means the world to him.
“It’s my brother’s dream though to play for Arsenal.
“Given all that Wilfried has given to Crystal Palace to help them remain a Premier League club, I hope Palace will be able to see their way to agreeing a deal with Arsenal that allows Wilfried to realise his dream of playing European football for the club he’s supported since childhood.”