Sane’s club confirmed on Monday that he was in Manchester, rather than joining them on a pre-season training camp, where he was expected to undergo a routine medical examination.
City are believed to have agreed to pay an initial £37m for one of German football’s brightest prospects, plus up to £5m in additional fees depending on his success in the Premier League.
Sane’s arrival would take City’s spending this summer past the £70m mark. They have already captured another Germany international, midfielder Ilkay Gundogan from Borussia Dortmund for £21m, and Spain forward Nolito, who cost £14m from Celta Vigo.
Everton and England defender John Stones is likely to be new manager Pep Guardiola's next signing in a £50m deal.
They have also been linked with Gundogan’s prolific former team-mate Pierre-Emerick Aubameyang, although Dortmund are thought to be unwilling to sell another key player, having also lost Henrikh Mkhitaryan to Manchester United and captain Mats Hummels to rivals Bayern Munich.
Sane, who was part of the Germany squad that reached the semi-finals of this summer’s European Championship, has scored 13 goals in 57 appearances for Schalke.