Speculation has raged around the Ivory Coast international for much of the summer.
The 26-year-old, who has starred for Palace since returning to Selhurt Park from a forgettable stint at Manchester United, has opened the door for a move as he is eager to step back up onto a European stage.
Everton are also said to be keen, with it suggested that another late bid for Zaha could be lodged by those at Goodison Park before the deadline passes.
That may be the only potential landing spot available to the highly-rated forward, with leading Premier League sides now looking elsewhere.
Zaha had been billed as a suitable target for Liverpool by some, while reigning champions City are forever in the market for players with match-winning creativity.
Reds legend Barnes has, however, claimed that the Eagles star is too individualistic to have been considered by Jurgen Klopp or Pep Guardiola.
“He likes to dribble around five defenders. Therefore, he can't go to Man City and Liverpool; he can't go wandering all over the field.
“He plays whatever position he wants for Crystal Palace, and picks up the ball on the right and the left – that's not fitting into a team.
“He's got great ability and that's why I think Crystal Palace suit him down to the ground because they allow him to do that, he won’t be allowed to at another club.
“I don't think a player of his ilk should be looking to be a Plan B player to come off the bench.”
Zaha enjoyed the most productive campaign of his career in 2018-19, recording 10 goals and as many assists in the Premier League, but that has still not been enough to earn him the high-profile switch he craves.