Rafael Benitez’s side are looking to sign a new right-back before the transfer window slams shut and have identified Patterson as a potential recruit.
Everton stepped up their interest in the 19-year-old by making an initial offer of £5m, but saw the bid immediately turned down by Rangers.
The Merseyside-based club then returned with an improved offer of £8m plus add-ons for the Scotland international.
However, the Glasgow giants, who rate Patterson significantly higher, were quick to turn Everton’s proposal down again.
Having seen two offers for Patterson rejected, Everton have indicated that they will not return with a third bid for the right-back.
With Everton indicating that they will not increase their offer, Rangers can rest easy that they will not have to be tested by any further bids for Patterson.
The development will be a boost for the Gers, who do not want to lose Patterson before the transfer window closes.
