Guillermo Cantu says he contacted Klopp through a third party and offered him the chance to manage El Tri but was told the German was intent on heading to England.
Instead, Klopp, who was ending a seven-season spell in charge of Borussia Dortmund, where he won back-to-back Bundesliga titles in 2010-11 and 2011-12, signed a deal at Liverpool in October 2015.
"I asked through a third party if he'd be interested in the Mexican national team and the message back was, 'I appreciate it, many thanks, but I still have a lot to do in club football and I want to go to England,'" said Cantu in an Instagram Live interview with TUDN.
After Klopp declined the job, Cantu went on to hire Juan Carlos Osorio as a manager in 2015, a decision that Cantu says he does not regret.