Pep Guardiola’s City and Jose Mourinho’s United are joint top of the table after five matches, with Klopp’s side five points off the pace in eighth.
The Liverpool manager takes his side to Leicester tomorrow looking for a quick upturn in fortunes after a run of four matches without a win.
That included a 5-0 thrashing at Manchester City and a disappointing 1-1 draw at home to Burnley, as well as a Carabao Cup exit at the King Power Stadium on Tuesday.
Klopp says that seven years in charge of Borussia Dortmund, where he was competing with Bayern Munich for trophies, taught him plenty about the pressures of expectation.
But he has acknowledged that those pressures have been raised at Liverpool, where the rivalry with Manchester as a city, and with United as a club, is huge.
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He said: “Everywhere it is the same but here it is a little bit more. When the Manchester teams are flying, that makes it even more difficult.
“But I cannot change this. What I can do is cool the situation here down and do the right things again and work on the other things. That is how it is always.”
Klopp’s side have looked threatening going forward this season but their defence has been vulnerable.
Yet the manager is adamant there is no need to panic as the club jostle for position in the title race.
He said: “You really think there would be one per cent of a reason I would panic?
“We have eight points. In four games, we were the clear better side but we didn’t get the results.
“Now we could panic because of not getting the results or we could say, ‘It’s still not good enough playing football’.
“Do you think any team in the world loves playing against us in this moment? Or do they hope, like Burnley, that they only have to go for set-pieces and get whatever they want and can concede 35 shots on target or whatever and it’s clear they will be lucky?
“I’m not in panic. I cannot imagine that anybody thinks about this.”