It is ludicrous to even think about firing Klopp. As Liverpool have made great strides under him and there's no clear cut better option. However, he is not beyond criticism (not that's what you're suggesting) if pool don't make the top 4... he bears the brunt of the blame IMO.
Have they though? They were closer to the league title under Rodgers in 2014 than they have been under Klopp so far (and they won't get that close this year either). Yes, they qualified for the CL last year but that isn't meant to be a one off, they intended for this year to be another step in the right direction, not a year of mediocrity.
I'm a big Klopp fan but aside from his numerous brilliant victories and the generally attractive style of play, the man has also overseen one of the more remarkable runs of futility by a major club in recent history. The half season that got him fired at Dortmund was basically a typical Liverpool loss under Klopp on a loop. How exactly did that happen? Maybe it's because when the defensive breakdowns start occurring frequently enough every team at almost every talent level can pull a number on a Klopp team.
This trouble could be seen brewing for a while and the transfer window over the off-season should have had alarm bells ringing among Liverpool fans. Liverpool simply do not have the personnel to play "Kloppball" with any sort of consistent success and the failure to shore it up over the summer all but guaranteed that this season would not be a step forward. I don't know who controls transfer dealings at Liverpool, but whoever it is basically has gotten the club into a pickle.
If by January/February they're still where they're in the league right now, not only would Klopp be on the 'hot seat', I'd find it hard to believe he'd survive a season without European qualification.