The Germany legend has guided the US into next summer's World Cup in Brazil after his side finished top of the CONCACAF Fourth Round of qualification.
Switzerland have also qualified for next summer's showpiece, after which current head coach Ottmar Hitzfeld, 65, is set to step down.
Former Spurs striker Klinsmann, who guided Germany to the semi-finals of the 2006 World Cup, is seen as a leading candidate for the Swiss job, as are Austria head coach Marcel Koller, Thorsten Fink and former Spurs boss Christian Gross.
While Klinsmann says several parties have enquired about his services after his deal with the US expires following the World Cup, he is not naming specific jobs
He told BILD: "I have been receiving several enquiries in recent months.
"Those enquiries were from Europe-based clubs and from several Football Associations as well.
"And all of them were about the time after the 2014 FIFA World Cup."