It was reported earlier in the month that Tottenham would listen to offers for Rose, amongst a list of other players, and that remains true up until Thursday's transfer deadline.
Rose criticised the club in 2017, citing the wage structure and their ambitions as the reasons, and despite apologising this it led to a rift between the Englishman and Pochettino.
In addition, with Ben Davies putting in a string of strong performances, he has struggled for playing time at White Hart Lane.
Since returning from a serious injury sustained in January 2017, Rose made only 10 appearances for Spurs in the league, although he still made Gareth Southgate's World Cup squad.
Schalke finished second in the Bundesliga last season, behind Bayern Munich, but still have weeks until their own transfer deadline.