The 29-year-old has recently been linked with a move away from Selhurst Park, but Jones is not expecting Mikel Arteta and Edu to go after him.
What is the latest news involving Zaha?
Rather than Arsenal, it looks like it is their London rivals Chelsea who are weighing up making an offer for Zaha.
As per the MailOnline, the Blues are considering a late bid for the Ivorian, with Thomas Tuchel still looking to improve his attack.
Zaha now has less than a year remaining on his contract, so Palace could be tempted to sell in order to avoid the risk of losing him for free next summer.
Jones has previously told GIVEMESPORT that the south London club would likely ask for up to £30m for their talisman. With it being so late in the transfer window, though, it remains to be seen if they are willing to play ball at all.
What has Jones said about Zaha and Arsenal?
According to Jones, Arsenal have been monitoring Zaha and there is one figure at the club who would like them to make a move for him. The journalist, though, does not expect that to happen in a boost to Chelsea.
Speaking to GMS, he said: “I think one thing that will encourage them [Chelsea] at this moment is that Arsenal haven’t made any progress with it. It seems people within the club are indicating that they’re not going to go for Zaha at this point.”
Why are Arsenal not pursuing Zaha?
While it is clear that Arteta wants another wide man at the Emirates following the failed pursuit of Raphinha, Pedro Neto seems to be the man he is eyeing.
Last week, the Evening Standard reported that Arsenal are still keen on signing a new winger and the Wolves star is a top target for them.
Ultimately, their interest in Neto explains why Zaha is not a priority for them right now. They may struggle to land the 22-year-old, though, with the aforementioned outlet claiming that the Gunners fear they will be priced out of a move for him.
Whatever the case, be it Zaha or Neto, Arsenal could do with the addition of a new wide player, especially with Nicolas Pepe out on loan for the season. As things stand, however, it does not look like they will sign either.