After winning the award for the first time, Klopp said to the Premier League's website: "I went into the office of my coaches and said, 'We are Manager of the Month! I don't need personal prizes but it's nice now I have another one'.
"We are on a good way but there are so many good teams in this league and so many difficult games for us, which we have to be ready for.
"I'm fine with the moment, but we have to play a few more games and hopefully we perform well."
It was an award double for Liverpool with Jordan Henderson receiving the goal of the month accolade for his strike against Chelsea.
Son, meanwhile, was named as the month's top player after scoring four times and registering an assist in Tottenham's three league matches – all of which were victories.
"This is my dream, to get the award," Son said. "I want to get it a second time, a third time.
"When I take the award home my family will be crazy like you can't believe - they will be really happy.
"I can't believe I'm the first player from Asia. Many players deserved it - Ji Sung-Park had great seasons.
"I'm very thankful for the support. They support every day, every weekend, every game."