The England international is being offered the chance to resurrect his career at the second placed Bundesliga side from last season, with the Germany club aiming to compete with any English club who come in for the full-back ahead of the transfer deadline on Thursday.
However, the transfer window in Germany remains open until August 31, which may help Schalke secure Rose on loan.
Schalke return to the Champions League this season for the first time since the 2014-15 campaign.
Despite being the subject of much transfer speculation last summer, including links to Manchester United, Rose has fallen below Ben Davies in the pecking order at Spurs and only made 10 appearances under Mauricio Pochettino last season.
Rose also drew controversy when he delivered an interview chastising Spurs' transfer policy and player compensation.
Despite his lack of playing time, the 28-year-old operated as Ashley Young's deputy in England's World Cup squad over the summer, which saw them finish as semi-finalists for the first time since 1990.
Toby Alderweireld's future is also in doubt ahead of transfer deadline day with Manchester United having agreed a deal for the defender but Spurs wanting a replacement in place before letting him leave.
Alderweireld, like Rose, saw himself fall down the pecking order under Pochettino with the London club.