The Bundesliga outfit were understood to have been interested in signing Loftus-Cheek on loan for the 2018-19 campaign, only for the 22-year-old to turn down a move in order to fight for his place in the Chelsea team.
The Daily Star reports that Schalke have now identified Drinkwater as a possible alternative to replace Max Meyer, who left the club earlier this summer.
New Chelsea boss Maurizio Sarri is willing to sanction the exit of Drinkwater, who made just five Premier League starts for the club last season following his arrival from Leicester City.
The 28-year-old was not involved in the matchday squad for Chelsea's 3-0 win over Huddersfield Town on the opening weekend of the new season and appears to be behind Jorginho, N'Golo Kante, Mateo Kovacic, Cesc Fabregas, Ross Barkley and Loftus-Cheek in the midfield pecking order at Stamford Bridge.
Drinkwater could still choose to fight for his place in the team, but Chelsea are open to sending the England international out on loan to a foreign team before the end of the European transfer window on August 31.