Klopp masterminded a superb second half to the Premier League season, yet it was not enough to overhaul City, who claimed the title on a dramatic final day of the campaign on Sunday.
The Reds went the entire season unbeaten at home, doing so for a joint-record fifth time in the Premier League.
Liverpool has also excelled in cup competitions under Klopp this term, having won the FA Cup and Carabao Cup.
It will complete a cup treble on Sunday (AEST) if it overcome Real Madrid in the Champions League final in Paris.
Reflecting on the dramatic final day, which saw City score three goals in the space of five minutes to beat Aston Villa 3-2, Klopp said: "It was a bit nervy, it wasn't the best outcome for us, but we are already over it.
"And when you win a prize like this you are either a genius or you have the best coaching staff in the world. I am here with four of my coaching staff, and they know how much I appreciate them."
It is the second time Klopp has claimed the award, which is combined with the Premier League Manager of the Year award, after the German was recognised in 2020 for leading Liverpool to the title.
Eddie Howe, Patrick Vieira and Thomas Frank, of Newcastle United, Crystal Palace and Brentford respectively, were also up for the award.