Schar, 33, is off contract at the end of the season and is free to sign pre-contract terms with Bundesliga clubs from January 1.
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The Sun says several German top-flight clubs are interested in the Switzerland international.
And it's suggested Newcastle could be willing to sell Schar in January to ease Profit and Sustainability concerns for Toon chiefs.
For the moment, Newcastle are yet to contact Schar's camp about new contract talks.