The 32-year-old has enjoyed a successful five-and-a-half-year spell with the Citizens since joining from Tottenham Hotspur in 2017, winning a total of 11 trophies including four Premier League titles.
Walker has made 229 appearances for City across all competitions, but his game time has been limited so far this season, starting only 12 of his 22 matches.
The England international missed a number of games due to injury before the World Cup, but since the tournament in Qatar, Walker has lost his regular place in Pep Guardiola's first XI, playing the full 90 minutes in only three of City's last nine matches.
Indeed, 18-year-old Rico Lewis has taken his chance in the first team and has excelled at right-back since the World Cup, with Walker forced to play second fiddle to the academy graduate as a result.
Walker has less than 18 months remaining on his contract at the Etihad Stadium and negotiations over an extension are yet to bear fruit.
According to Turkish news outlet ASpor, Fenerbahce head coach Jorge Jesus has given the club the green light to pursue a move for Walker this month.
The report adds that the Yellow Canaries are ready to step up their interest and make an offer in the region of €6m (£5.3m) to sign the right-back, who is said to be keen to 'embark on a new adventure'.
Fenerbahce will not need to rush their potential pursuit of Walker as the Turkish transfer window closes on February 8, eight days after the English window shuts.
It remains to be seen whether City would approve such a transfer having already agreed to loan fellow full-back Joao Cancelo to Bayern Munich for the rest of the season amid a bust-up with Guardiola.
Walker, a 73-cap England international, could become the second English player after Dele Alli to move to the Turkish top flight this season.
Alli moved to Besiktas on loan from Everton in the summer, but the attacking midfielder has endured a difficult time in Istanbul and has received scathing criticism from his own manager Senol Gunes.
The likes of former Spurs duo Aaron Lennon and Steven Caulker and former Liverpool, Chelsea and Man City striker Daniel Sturridge have also played in Turkey in recent years for Kayserispor and Trabzonspor respectively.
Fenerbahce currently sit second in the Super Lig table, four points behind league leaders Galatasaray, and are bidding to win their 20th top-flight title this season.