The Bundesliga club announced on Monday that the midfielder was in Manchester and had not joined their pre-season training camp in Austria.
City manager Pep Guardiola confirmed on 21 July he wanted to sign Sane.
The 20-year-old helped Germany reach the semi-finals at Euro 2016, having made his international debut in November.
He told Schalke sporting director Christian Heidel earlier this summer that he wanted to leave the club, according to German media.
Sane, a product of the youth system at Schalke, made his senior debut for them in April 2014.
He scored eight goals in 33 Bundesliga appearances last season as Schalke finished fifth, missing out on Champions League qualification for the second year in a row.
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Leroy Sane is close to completing a move to Manchester City from Schalke for a fee of £37m plus add-ons.