The Crystal Palace winger is known to be a target for the 'Blues' as they look to build a squad capable of challenging in the upper echelons of the division once again.
According to the aforementioned source, the 25-year-old is open to a move to a big side, though Palace's £70m valuation of the Ivorian is putting interested parties off making a swoop.
Chelsea bids to date are thought to have amounted to little more than £30m with several players on loan, making a deal look unlikely at this stage.
Zaha only signed a new long-term deal with the club last year, but has rejected improved terms this summer as the club looked to further protect their asset.
That decision to snub yet another new deal has led some to suggest he has his sights set on a departure, but any move to Chelsea is likely to rely on Willian moving on before the deadline, with Manchester United thought to retain a serious interest in the Brazilian.