It says the 20-year-old will agree a four-year deal with City in a matter of days to start in time for the next Premier League season.
His contract with Schalke was set to last until 2019.
City has been on a spending spree during the summer transfer window, with new manager Pep Guardiola already signing Spanish star Nolito - as well as Ukraine winger Oleksandr Zinchenkod and FC Ufa's Alexander Zinchenko.
Sané is currently in Los Angeles, taking a break after Euro 2012 - which saw Germany go out to France in the semi-finals - but he has already told friends that a City move is happening, accoring to the Daily Mail. Guardolia's presence is said to be a major draw for Sané, who did not want to wait for negotiations to run into next year.
As a midfielder he has scored 11 goals in 47 league appearances for the Shalke, and is known for being an intelligent playmaker who can work as well in attack or defence mode.
Manchester United's prospects for next season look strong, with bookmakers pegging the club as the favourite to win the Premier League. It scraped fourth place in the table last season - well below expectations but enough to ensure Champions League action, which will be a big draw to future transfers.