Citing sources, Sky Sports said City boss Pep Guardiola is confident of signing Walker and hopes to secure his services for a fee of around £40m ($51m). Spurs are reported to value the right-back at £50m.
City are expected to submit a formal bid to Spurs after England's friendly against France in Paris on 13 June.
Walker has a contract with Spurs until 2021 that is worth £70,000 per week. Sources told Sky that the England international was relaxed over his future and is not agitating for a move away from the north London club.
However, the 27-year-old is understood to be unhappy at not being selected for some matches towards the end of the season, with Spurs boss Mauricio Pochettino occasionally opting for Kieran Trippier at right-back.
Walker told Sky Sports on 7 June that he was not thinking about his future and that he was enjoying his rivalry with Trippier.
"[It's not a distraction] – not at all. Coming away from club football gives you head space. I'm an England player now and we've got to concentrate on these games," he said.
"[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.
"I don't ever say a bad word or wish bad on him because it's more about the team rather than individual performances. We're putting on our England shirts now and I'm sure, whoever gets the nod, the other will be behind him.
"I've known Trip since U19s. We had a tournament with England in Ukraine. I played centre-half and he played right-back I believe, and it was a friendly rivalry. We're two northern boys down south and we bounce off each other. We get on really well," Walker added.