The French defender has been a key part of Carlo Ancelotti's side this season with the Toffees currently sitting in the top-four going into Christmas Day.
Sportsmail revealed last week that Pep Guardiola's City are keen on boosting their left-back options for the second half of the season and Digne has been named as one of the options.
But according to The Times, Everton are looking to tie the 27-year-old to a new contract and convince him that his career is best spent at Goodison Park.
The full-back is currently out with an ankle ligament problem that he picked up in training at the end of November.
It is expected Digne will miss another four weeks of action which means he may be absent for most of Everton's January fixtures.
The Frenchman had started all but one of Everton's matches this season up to that point and while he is yet to score, he has contributed with four league assists.
French international Digne is closing in on 100 appearances in all competitions for Everton since his £18million arrival from Barcelona in the summer of 2018.
The left-back scored four Premier League goals in his debut season to immediately endear himself to the Toffees faithful.
A new contract at Finch Farm would be a blow for Pep Guardiola and Manchester City who are trying to acquire a new left-back with Benjamin Mendy still struggling for consistency in terms of fitness.
Ajax left-back Nicolas Tagliafico is another player City are interested in signing this winter. The Argentinian has made 17 appearances in all competitions so far this season for the Dutch giants.
City are also looking at bringing in a striker with both Sergio Aguero and Gabriel Jesus suffering injury problems this season.
Benfica striker Darwin Nunez has been earmarked as a potential option to boost Guardiola's side and their title hopes.
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