The Independent says City boss Pep Guardiola's pursuit of Walker is set to drag on into next week as a result of Spurs refusing to budge on their valuation of a player who has four years left on his contract.
Walker, 27, reportedly wants to quit Tottenham for City after being left out of the Spurs team by manager Mauricio Pochettino in crucial fixtures towards the end of last season.
He is also aware that he could double his current weekly wages of around £70,000 if he moves to the Etihad Stadium.
City are also pursuing a move for Juventus right-back Dani Alves, although the acquisition of the Brazilian would not necessarily lead to them dropping their interest in Walker.
Walker is reported to have clashed with Pochettino over fitness and training and was unhappy at being left out of the team for key matches such as the FA Cup semi-final against Chelsea and the north London derby against Arsenal.
Spurs already have a ready-made replacement for Walker in Kieran Trippier, while the club are also reported to be monitoring Porto right-back Ricardo Pereira.
Walker told Sky Sports before England's games against Scotland and France earlier this month that he was not thinking about his future and that he was enjoying his rivalry with Trippier.
"[The rivalry] has pushed us both on as players, most definitely. We get the best out of each other. Even if Trips plays, or I play, we've said privately to each other than we'll will each other on," he said.
"I don't ever say a bad word or wish bad on him because it's more about the team rather than individual performances."